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The Interaction of Light and Matter. Screen. Metal plate. Wire. Time. Learning Objectives. Problems with Bohr’s Semiclassical Model of the Atom - PowerPoint PPT Presentation
Citation preview
The Interaction of Light and Matter
Time
Screen
Wire
Metal plate
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
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
Learning Objectives Paulirsquos Exclusion Principle
No two electrons (particles) can share the same quantum state Relativistic 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
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
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
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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 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 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
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
Learning Objectives Paulirsquos Exclusion Principle
No two electrons (particles) can share the same quantum state Relativistic 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
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
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
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
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
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
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
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
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 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 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
Learning Objectives Paulirsquos Exclusion Principle
No two electrons (particles) can share the same quantum state Relativistic 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
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
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
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
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
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
Learning Objectives Paulirsquos Exclusion Principle
No two electrons (particles) can share the same quantum state Relativistic 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
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
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
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
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
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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