The Interaction of Light and Matter

Time

Screen

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Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical system

Quantum mechanical atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics -why is the angular momentum

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

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Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

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In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

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In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical system

Quantum mechanical atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics -why is the angular momentum

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics -why is the angular momentum

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 106

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics -why is the angular momentum

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics -why is the angular momentum

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 103
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• Slide 105
• Slide 106

Bohrrsquos Semiclassical Atom Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another What are the two problems in Bohrrsquos model of the atom that cannot be explained

using classical physics- why is the angular momentum of the electron quantized- why does an electron in a permitted orbit not emit electromagnetic waves as demanded by

Maxwellrsquos equations

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 106

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical cat (Schroumldingerrsquos cat)

Quantum mechanical atom

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

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Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 105
• Slide 106

Wave-Particle Duality Recall the wave-particle duality of light

Light ndash in the form of electromagnetic waves ndash shows its wave-like properties as it propagates through space

Light ndash in the form of photons ndash shows its particle-like properties as it interacts with matter

If electromagnetic waves ndash light ndash can exhibit particle-like properties can particles (eg protons electrons neutrons atoms molecules bacteria plants animals humans planets stars galaxies) exhibit wave-like properties

Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Wave-Particle Duality

Louis de Broglie 1892-1987

This revolutionary idea for the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis

Recall that in 1905 Einstein used Planckrsquos idea that the energy of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon to explain the photoelectric effect

Recall that from the Theory of Special Relativity the energy and momentum of a photon are related by

as had been verified in 1922 by Compton through the Compton effect

Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Wave-Particle Duality Purely from symmetry arguments de Broglie proposed that all matter (from

elementary particles to people planets stars and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by

The wavelength given by Eq (517) is known as the de Broglie wavelength

In this view the wave-particle duality applies to everything in the physical world- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality If everything ndash including us ndash propagate as waves should we worry about being

diffracted when walking through the door

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting Hint Conditions for destructive interference

Notice also that the bright fringes in the interference pattern become dimmer for larger θ

L raquo D

for m = 1 2 3 hellip

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality So our wave-like properties have very short wavelengths (depending on our mass

and speed) Why then do we not have to worry about diffracting The maxima in the interference pattern is concentrated at θ = 0o ie we walk through and appear behind the door in the direction of motion at the door

L raquo D

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality In 1927 Clinton J Davisson and Lester H Germer directed a beam of electrons

on a highly polished single crystal of nickel The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0215 nm about the size of an atom

The electron beam they used had an energy of 54 eV corresponding to a de Broglie wavelength of 0167 nm The first-order (m = 1) maximum should therefore occur at = sin-1 (d) = 51o in agreement with their measurements

Clinton J Davisson (1881-1958 left) and

Lester H Germer(1896-1971 right)

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

Wire

Metal plate

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

How do electrons interact with the screen As particles which is why individual points appear on the screen (where electrons with strike ndash interact with ndash the screen)

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons behave purely like particles Two stripes

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties

Wire

Metal plate

Screen

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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• Slide 105
• Slide 106

Wave-Particle Duality Modern experiments such as that performed in 1989 by A Tonomura and his

colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Joumlnssonrsquos experiment in 1961) is equivalent to the double-slit experiment for light About 10 electrons are emitted a second each accelerated to ~04 c

What image would you see if electrons propagate with wave-like properties Interference pattern

Each electron passes through both slits and interferes with itself thus revealing itrsquos wave-like properties (Obviously a particle can only go through one slit)

Wire

Metal plate

Screen

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality

>

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as

atoms

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality Instead of double slits experiments using just one slit also have been performed

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality Interference pattern reported by Zeilinger and his collaborators for an experiment

where they directed a beam of neutrons through a single slit

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality Recall that Bohrrsquos model of the atom combined both classical physics and

quantum mechanics and has as its two central ingredients- electrons in circular orbits around the the nucleus (classical physics)- the orbital angular momenta of electrons are quantized (quantum

mechanics) Bohr also assumed that electrons only absorb or emit electromagnetic waves when they make make a transition from one

permitted orbit to another If electrons behave as waves can you explain

why the angular momentum of electrons are quantized (can only have certain permitted orbits)

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohrrsquos

model of the atom is simply a manifestation of the wave-like nature of the electron The circumference of an electronrsquos orbit must be equal to an integral number of wavelengths (ie integral number of de Broglie wavelengths) for the electron to undergo constructive interference Otherwise the electron will find itself out of phase and suffer destructive interference

Based on this consideration one can show that electron can only have angular momenta given by

Assignment question

In this description we no longer envisage electrons as particles orbiting at different permitted radii from the nucleus in an atom but as standing waves surrounding the nucleus in an atom

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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• Slide 106

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons can orbit an atom without emitting electromagnetic radiation Electrons are not orbiting as particles interacting through Coulomb forces with their nuclei but propagate as waves

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
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Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can absorb electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When absorbing electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only absorb photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 105
• Slide 106

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

Can you now explain why electrons in atoms can emit electromagnetic radiation but only at certain wavelengths

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 105
• Slide 106

Wave-Particle Duality Now using the principles for the wave-particle duality of the physical world

- everything exhibits its wave properties in its propagation- everything manifests its particle nature in its interactions

When emitting electromagnetic radiation an electron a photon and the atomic nucleus interact as particles Electrons can only emit photons at wavelengths that change their de Broglie wavelengths by certain fixed amounts

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 106

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Probability Waves Quantum mechanics describe particles in terms of probability waves Consider a particle that comprises the following probability wave Ψ a sine wave

with a precise wavelength propagating along the x-direction

The momentum p = h of a particle described by such a wave is known precisely (as the wavelength is known precisely)

The probability of finding the particle at a given location x is given by

P(x) = = [0 ei(kx-t)] [0 e-i(kx-t)]

= |02|

which is a constant independent of x or t Thus the particle can be found with equal probability at any point along the x-direction its position is perfectly uncertain ie a sinusoidal wave has no beginning or end

Probability wave Ψ

(xt) = 0 ei(kx-t)

where k = 2 = 2 ν

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

Probability wave Ψ

x

Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Probability Waves Consider now a particle that has a probability wave Ψ that is equal to the

addition of several sine waves with different wavelengths

The position of such a particle can be determined with a greater certainty because P(x) = is large only for a narrow range of locations

On the other hand because Ψ is now a combination of waves of various wavelengths the particlersquos momentum p = h is less certain

This is naturersquos intrinsic tradeoff the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp is inversely related As one decreases the other must increase This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature

Probability wave Ψ

x

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How do you explain this if electrons behave like waves Hint how can you precisely control thethe position of an electron and what are the consequences of doing this

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 105
• Slide 106

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

How do you prevent the electron beam from producing an interference pattern By precisely controlling the position of the electrons so that they only enter one slit If we can be sure that an electron only passes through one slit we no longer see an (double-slit) interference pattern

How can you precisely control thethe position of an electron and what are the consequences of doing this By using electrical or magnetic deflecting plates But these deflecting plates change the momentum of the electrons (by accelerating electrons to the desired direction) and so you lost accurate control of the momentum and hence wavelength of these electrons Electron waves at difference wavelengths produce interference patterns with different locations for their maxima and minima hence an (double-slit) interference pattern is lost

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 104
• Slide 105
• Slide 106

Probability Waves and the Two-Slit Experiment As an illustration of naturersquos intrinsic tradeoff between the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δp consider the following

Conversely if you try to precisely control the momentum of an electron you will lose good control of its position As a consequence an electron can go through both slits (obviously an electron can go through both slits only if it behaves like a wave) and interfere with itself to produce an (double-slit) interference pattern (I leave it up to you to imagine how you could try to precisely control the momentum of an electron It is not trivial)

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 106

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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• Slide 106

Probability Waves and the Two-Slit Experiment In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Probability Waves and the Two-Slit Experiment

Wire

Metal plate

Screen

In the double-slit experiment for electrons why is the minima in the interference pattern not perfectly dark Because the momentum and hence wavelength of the electrons cannot be perfectly controlled

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Probability Waves and the Two-Slit Experiment Notice that the mimina in the double-slit interference pattern for neutrons is not zero

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 105
• Slide 106

Probability Waves and the Single-Slit Experiment Notice that the mimina in the single-slit interference pattern for neutrons is not zero

Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Heisenbergrsquos Uncertainty Principle In 1927 the German physicist Werner Heisenberg presented the theoretical framework for the inherent ldquofuzzinessrdquo of nature showing that the uncertainty in a particlersquos position Δx and the uncertainty in its momentum Δt is related by

He also showed that the uncertainty of energy measurement ΔE and the time interval over which this measurement is taken Δt is related by

Recall that = = 1054571596(82) x 10-34 J s

Werner Heisenberg 1901-1976

Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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Heisenbergrsquos Uncertainty Principle Heisenberg based his arguments on the principles of classical optics through a thought experiment known as Heisenbergrsquos microscope

Heisenberg begins by supposing that an electron is like a classical particle moving in the x direction along a line below the microscope as in the illustration to the right Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle ε with the electron Let λ be the wavelength of the light rays Then according to the laws of classical optics the microscope can only resolve the position of the electron up to an accuracy of

This equation (definition of resolution) is analogous to the alternative definition for the angular resolution of a telescope with a diameter D

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Heisenbergrsquos Uncertainty Principle When an observer perceives an image of the particle its because the light rays strike the particle and bounce back through the microscope to their eye When a photon strikes an electron the latter recoils (recall the Compton effect) with a momentum proportional to hλ where h is Plancks constant It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment He writes that the recoil cannot be exactly known since the direction of the scattered photon is undetermined within the bundle of rays entering the microscoperdquo The electrons momentum in the x direction is only determined up to

Combining the relations for Δx and Δp we have

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Measurement Precision This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to

make an improved image of BetelgeuseWe target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the

bandwidth of each of eight images that we plan to make over the frequency range 40ndash48 GHz) with robust weighting about two orders of magnitude higher than that previously attained by Lim et al (1998) with the VLA as shown in Fig 1 The required integration time is about 3 hrs With an estimated observing efficiency of 60 (including overheads for absolute flux calibration and pointing checks) we will require a total observing time of 5 hrs

Using Eq (520) explain why observers sometimes integrate for hours when observing astronomical objects

As the time available for an energy measurement increases the inherent uncertainty in the result decreases Thus making more precise measurements require longer observing times

Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Measurement Precision Apart from the identified absorption lines as indicated by the yellow line much of

of the remaining fluctuations in the measurements (black solid line) reflect the uncertainty in the intensity measurements at a given wavelength

Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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Quantum Mechanical Tunneling Because of the inherent uncertainty between a particlersquos momentum and position a particle can penetrate a barrier even though the particlersquos energy is lower than the barrier potential

If you know the energy of the particle precisely (and it is smaller than the barrier potential) you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small)

So a (small) fraction of particles on one side of a barrier can tunnel through to the other side an effect called quantum mechanical tunneling

Probability wave Ψ

x

Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
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Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as water or light waves When such a wave enters a medium through which it cannot propagate its amplitude decays with distance in this medium the wave becomes evanescent (fades away)

If the barrier width is sufficiently small the amplitude of the wave may not decay away completely before reaching the other side of the barrier where the wave can once again propagate

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion

The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon

For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together their required kinetic energy corresponds to a gas temperature of ~1010 K

Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 157 times 107 K Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (ie particle speeds are distributed according to the Maxwell-Boltzmann distribution) not enough protons can overcome their Coulomb repulsion to produce the Sunrsquos observed luminosity

Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbits as prescribed in Bohrrsquos model of the atom Rather the electron orbits must be imagined as fuzzy clouds of probability with the clouds being mode ldquodenserdquo in regions where the electron is more likely to be found

Bohrrsquos model for the hydrogen atom

Modified Bohrrsquos model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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• Slide 106

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom

Quantum mechanical model for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 93
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• Slide 95
• Slide 96
• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Wavelengths closer to the de Broglie wavelengths interfere more constructively and so there is a higher probability of finding electrons closer to the de Broglie wavelengths Quantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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Electronic Orbitals in Atoms Heisenbergrsquos uncertainty principle implies that electrons cannot have well defined

orbital angular momenta as prescribed in Bohrrsquos model of the atom How do we explain this concept in terms of the de Broglie wavelengths for electrons in an atom Conversely wavelengths further from the de Broglie wavelengths interfere more destructively and so there is a lower probability of finding electrons further from the de Broglie wavelengthsQuantum mechanical model

for the hydrogen atom

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Electronic Orbitals in Atoms One of the predictions of such a model is that spectral lines cannot be infinitely

sharp but must have a certain width (natural linewidth) as is indeed observed Natural line profile is a Lorentzian profile

Modified Bohrrsquos model for the hydrogen atom

Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Learning Objectives Problems with Bohrrsquos Semiclassical Model of the Atom Wave-Particle Duality

Everything exhibits its wave properties in its propagation and manifests its particle nature in its interactions

de Brogliersquos wavelength for electrons in an atom Wave-like Description of the Physical World

Probability waves in quantum mechanicsHeisenbergrsquos uncertainty principle

Some applications of Heisenbergrsquos Uncertainty Principle in ScienceAstrophysics Measurement precision

Quantum mechanical tunneling and nuclear fusion in starsNatural widths of spectral lines

Schroumldingerrsquos Wave EquationSolutions describe possible quantum states of a physical systemQuantum mechanical atom

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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• Slide 106

Wave-Particle Duality Although to be able to relate to the familiar we imagine an atom as electrons

orbiting a nucleus a proper description of an atom requires describing electrons as waves

Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
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Schroumldingerrsquos Equation Motivated to find a proper wave equation for the electron in 1926 the Austrian physicist Erwin Schroumldinger formulated the wave equation

now known as Schroumldingerrsquos equation that describes how the quantum state of a physical system (elementary particles atoms molecules etc) changes in time

Erwin Schroumldinger 1887-1961

Wavefunctions Ψ(x t) that satisfy Schroumldingerrsquos equation describe possible quantum states of a physical system Eg wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of the particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos equation is to quantum mechanics what Newtonrsquos equations are to classical physics (mechanics)

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 106

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Schroumldingerrsquos Equation Wavefunctions that satisfy Schroumldingerrsquos equation for a single particle describe the allowed values of a particlersquos energy momentum etc as well as its propagation through space

Plot of the real part of a possible wavefunction for a particle moving at a constant velocity

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 106

Quantum Mechanical Atom Schroumldingerrsquos wave equation

can be solved analytically for the hydrogen atom giving exactly the same set of allowed energies as those obtained by Bohr In addition to the principal quantum number n Schroumldinger found that two other quantum numbers and are required for a complete description of the electron orbitals such that the orbital angular momentum of the electron

where an is the angular momentum quantum number and n is the principal quantum number For historical reasons related to how spectral lines were first designated = 0 1 2 3 4 5 etc are referred to as s p d g f h etc

Erwin Schroumldinger 1887-1961

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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• Slide 106

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

For a single-electron atom such as hydrogen the different angular momentum quantum numbers with the same principal quantum number n have the same energy and are said to be degenerate (Note that this is not the case for a multi-electron atom)

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

states (n ) in a hydrogen atom Each orbital has a

characteristic shape reflecting the motion of the electron in that particular orbital this motion being characterised by an angular momentum that reflects the angular velocity of the electron moving in its orbital

f g

ml = 0

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 50
• Slide 51
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• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
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• Slide 77
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• Slide 84
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• Slide 86
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• Slide 92
• Slide 93
• Slide 94
• Slide 95
• Slide 96
• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Quantum States of the Hydrogen Atom The energy diagram for the hydrogen atom plotted in columns of constant orbital angular momentum quantum number

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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• Slide 103
• Slide 104
• Slide 105
• Slide 106

Quantum States of the Helium Atom The energy diagram for the helium atom plotted in columns of constant orbital angular momentum quantum number For a multi-electron atom like helium interactions between electrons result in atoms with the outermost electron having the same principal quantum number but different angular momentum quantum numbers having different energies

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 95
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• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

The projection of the orbital angular momentum in a specified direction (z-axis) the angular momentum vector component is also referred to as the magnetic quantum number The z-component of the angular momentum vector Lz can only have values with equal to any of the integers between and inclusive Thus the angular momentum vector can point in different directions

Eg for n = 1 = 0 and = 0 for n = 2 = 0 1 and = minus1 0 +1 for n = 3 = 0 1 2 and = minus2 minus1 0 +1 +2

Quantum Mechanical Atom

Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
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Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom

In the absence of any preferred direction in space (eg as defined by an electric or magnetic field) different orbitals with the same principal quantum number n have the same energy and are said to be degenerate

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Quantum Mechanical Atom Cross-section in xz-plane of the probability densities for an electron in different

excited states (n ) in a hydrogen atom

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
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• Slide 86
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• Slide 89
• Slide 90
• Slide 91
• Slide 92
• Slide 93
• Slide 94
• Slide 95
• Slide 96
• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

The Zeeman Effect An electron in an atom will feel the effect

of a magnetic field the magnitude of this effect depends on the electronrsquos orbital motion (ie magnitude and orientation of the electronrsquos orbital angular momentum through the magnetic quantum number

) and magnetic field strength B Electron orbitals with the same n and

but different values therefore have (slightly) different energies The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect after the Dutch physicist Pieter Zeeman

In the example shown the three frequencies of the split line are given by

The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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The Zeeman Effect Zeeman splitting of the Iron line as observed for a sunspot The Zeeman effect provides the only direct measure of magnetic field strengths in

astrophysics (There are several indirect methods to estimate magnetic field strengths)

λ

spat

ial d

imen

sion

al

ong

slit

slit

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 103
• Slide 104
• Slide 105
• Slide 106

Anomalous Zeeman Effect More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen usually involving even number of unequally spaced spectral lines This effect is called the anomalous Zeeman effect

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 42
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• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
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• Slide 84
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• Slide 88
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• Slide 90
• Slide 91
• Slide 92
• Slide 93
• Slide 94
• Slide 95
• Slide 96
• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Wolfgang Pauli 1869-1955

Based on the empirical knowledge of the properties of atoms (eg from their spectra) in 1925 the Austrian theoretical physicist Wolfgang Pauli proposed that electrons in an atom cannot share the same quantum state provided that each electron state is defined by four quantum numbers This rule which at the time did not have a theoretical basis is now known as the Pauli exclusion principle

Recall that three quantum numbers were known at the time the principal quantum number n the angular momentum quantum number and the magnetic quantum number The new quantum number that Pauli introduced could take on two possible values

Pauli Exclusion Principle

Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
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Quantum Mechanical Atom In 1925 George Uhlenbeck Samuel Goudsmit and Ralph Kronig associated the new quantum number introduced by Paul with the spin of the electron The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect

The spin angular momentum S is a vector of constant magnitude

with a z-component The only values for the electron spin quantum number ms are +frac12 or minusfrac12

Proton

Electron

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
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• Slide 105
• Slide 106

Quantum States of the Helium Atom Paraheliumorthohelium corresponds to helium atoms with their two electrons having antiparallelparallel spins

In orthohelium one electron is in the 1s state That state is not shown because the second electron cannot decay to the 1s state

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field

Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
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Quantum Mechanical Atom According to Maxwellrsquos equation a moving charge generates a magnetic field An orbiting electron therefore generates a magnetic field

A spinning charged sphere is an electrical current which according to Maxwellrsquos equations generates a magnetic field By analogy a ldquospinningrdquo electron generates a magnetic field

As a consequence the magnetic field generated by the electron due to its spin interacts with the magnetic field generated by the electron due to its orbital motion This effect is called spin-orbit coupling

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Quantum Mechanical Atom When the spectral lines of hydrogen are examined at very high spectral resolution they are found to be closely-spaced doublets The lines are split due to spin-orbit coupling of the electron and are known as fine structure lines

(The reason why the = 1 but not = 0 angular momentum quantum number is split is beyond the scope of this course)

0016 nm

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
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• Slide 44
• Slide 45
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• Slide 47
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• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
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• Slide 86
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• Slide 89
• Slide 90
• Slide 91
• Slide 92
• Slide 93
• Slide 94
• Slide 95
• Slide 96
• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines)

Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Recall that Fraunhofer determined that the wavelength of one prominent dark line in the Sunrsquos spectrum corresponds to the wavelength of yellow light emitted when salt is sprinkled in a flame

The sodium (D) line is actually a doublet (two closely-spaced lines) and is caused by splitting of a single spectral line into two by spin-orbit coupling

Quantum States of the Sodium Atom

ms = minusfrac12

ms = +frac12

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
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• Slide 8
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• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

The anomalous Zeeman effect usually involving the splitting of a spectral line into an even number of unequally spaced spectral lines in the presence of magnetic field is the same as the Zeeman effect but acting on lines which are split due to spin-orbit coupling

Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

The anomalous Zeeman effect is the same as the Zeeman effect but acting on spectral lines that are split by spin-orbit coupling The number of energy levels that results from the application of a magnetic field is beyond the scope of this course

Quantum Mechanical Atom

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 35
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• Slide 37
• Slide 38
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• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
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• Slide 63
• Slide 64
• Slide 65
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• Slide 91
• Slide 92
• Slide 93
• Slide 94
• Slide 95
• Slide 96
• Slide 97
• Slide 98
• Slide 99
• Slide 100
• Slide 101
• Slide 102
• Slide 103
• Slide 104
• Slide 105
• Slide 106

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Relativistic Schroumldinger Equation In 1928 the English physicist Paul Dirac combined Schroumldingerrsquos equation with Einsteinrsquos theory of special relativity to produce a relativistic wave equation for the electron

Diracrsquos solution -naturally included the spin of the electron- naturally explained Paulirsquos exclusion principle as being applicable to all

particles with spin of an odd integer times (such as electrons protons and neutrons) known collectively as fermions

- particles (such as photons) that have an integral spin do not obey Paulirsquos exclusion principle and are known as bosons

- predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic

moments)

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins (spin-spin coupling) The transition between these two states emits a photon with a wavelength of 21 cm

Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Quantum Mechanical Atom In 1933 Otto Stern and Walther Gerlach measured the effect of nuclear spin

For hydrogen atoms the nuclear spin quantum number can only have values of +frac12 or minusfrac12 Hydrogen atoms with parallel proton and electron spins have higher energies than those with antiparallel proton and electron spins The transition between these two states in the ground (n = 1) state emits a photon at a wavelength of 21 cm

spin-orbit coupling spin-spin coupling

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
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Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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• Slide 106

Hydrogen 21-cm Line The 21-cm line is one of the most important spectral lines in astronomy and

permits astronomers to map the distribution and bulk motion of neutral (unionized) atomic hydrogen gas in galaxies

Intensity of 21-cm line Velocity of 21-cm line

Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

• The Interaction of Light and Matter
• Slide 2
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Learning Objectives Paulirsquos Exclusion Principle

No two electrons (particles) can share the same quantum state Relativistic Wave Equation

Solutions to relativistic wave equation for an atom Complex Spectra of Atoms

Multielectron atoms with different ionization statesPermitted and Non-Permitted Transitions

Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

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Complex Spectra of Atoms In summary an electron in an atom is described by four quantum numbers

- principal quantum number n- orbital quantum number - magnetic quantum number = hellip- spin quantum number ms = plusmnfrac12

The electron can interact with itself through spin-orbit coupling The nucleus also has a spin quantum number The electron can interact with the

nucleus through spin-spin coupling In an atomion with a single electron obviously the electron does not interact

with other electrons The spectrum of such an atomion is hydrogen-like In a multielectron atom electrons interact not only with themselves and the

nucleus but also with each other (described as orbit-orbit and spin-spin coupling) The spectrum of multi-electron atoms therefore increases rapidly in complexity with the total number of electrons

Furthermore multielectron atoms with different ionization states (eg O I O II etc) have different spectra

Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

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Complex Spectra of Atoms Finally different transitions have different likelihoods of occurring Transitions that have a high

likelihood of occurring are known as permitted transitions and the resulting spectral lines known as permitted lines The lifetime of an electron at a permitted transition is 1 ≪s

Transitions that have a low likelihood of occurring are known as non-permitted or forbidden transitions and the resulting spectral lines known as forbidden lines The lifetime of an electron at a forbidden transition is gt1 s

Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

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Quantum Mechanical Atom An example of a forbidden transition is the transition of the hydrogen atom that produces the 21-cm line

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

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Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why

Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

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Complex Spectra of Atoms At optical wavelengths forbidden transitions are indicated by enclosing square

brackets eg [O III] These lines are not seen from gas even under the best vacuum conditions on the Earth (hence their designations as forbidden lines) but are only seen from gas in space Why Forbidden transitions have a relatively low probability of occurring or equivalently relatively long lifetimes At relatively high densities collisions between atoms (or ions) occur on timescales much shorter than the lifetimes of forbidden transitions the de-excitation energy is transferred to the kinetic energy of the colliding atom (or ion) At relatively low densities (as in interstellar space) collisions between atoms (or ions) occur on timescales much longer than the lifetimes of forbidden transitions thus permitting forbidden transitions that produce photons to occur

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