PHYSICS 460 Quantum Mechanics

PHYSICS 460 Quantum Mechanics

Prof. Norbert Neumeister Department of Physics

Purdue University

Fall 2013 http://www.physics.purdue.edu/phys460

Course Format •  Lectures:

–  Time: Tuesday, Thursday 10:30 – 11:45 –  Lecture Room: PHYS 331 –  Instructor: Prof. N. Neumeister –  Office hours: Tuesday 2:00 – 3:00 PM (or by appointment) –  Office: PHYS 374 –  Phone: 49-45198 –  Email: [email protected] (please use subject: PHYS 460)

•  Grader: –  Name: Tingting Shen –  Office: PHYS 221 –  Phone: 765-637-6923 –  Email: [email protected] –  Office hours: Monday: 11:00 am – 1:00 pm,

Friday: 1:00 pm – 3:00 pm

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Textbook The textbook is: Introduction to Quantum Mechanics, David J. Griffiths, 2nd edition We will follow the textbook quite closely, and you are strongly encouraged to get a copy.

Additional references:

•  R.P. Feynman, R.B. Leighton and M. Sands: The Feynman Lectures on Physics, Vol. III •  B.H. Brandsen and C.J. Joachain: Introduction To Quantum Mechanics •  S. Gasiorowicz: Quantum Physics •  R. Shankar: Principles Of Quantum Mechanics, 2nd edition •  C. Cohen-Tannoudji, B. Diu and F. Laloë: Quantum Mechanics, Vol. 1 and 2 •  P.A.M. Dirac: The Principles Of Quantum Mechanics •  E. Merzbacher: Quantum Mechanics •  A. Messiah: Quantum Mechanics, Vol. 1 and 2 •  J.J. Sakurai: Modern Quantum Mechanics

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A few recommended Books B. H. Bransden and C. J. Joachain, Quantum Mechanics, (2nd edition, Pearson, 2000). Classic text covers core elements of advanced quantum mechanics; strong on atomic physics.

S. Gasiorowicz, Quantum Physics, (3rd edition, Wiley, 2003). Excellent text covers material at approximately right level; but published text omits some topics which we address.

K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction, (Oxford University Press, 2009). This is a new text which includes some entertaining new topics within an old field.

L. D. Landau and L. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Volume 3, (Butterworth-Heinemann, 3rd edition, 1981). Classic text which covers core topics at a level that reaches beyond the ambitions of this course.

F. Schwabl, Quantum Mechanics, (Springer, 4th edition, 2007). Very good text for majority of course.

A few (random but recommended) books

B. H. Bransden and C. J. Joachain, Quantum Mechanics, (2ndedition, Pearson, 2000). Classic text covers core elements ofadvanced quantum mechanics; strong on atomic physics.

S. Gasiorowicz, Quantum Physics, (2nd edn. Wiley 1996, 3rdedition, Wiley, 2003). Excellent text covers material atapproximately right level; but published text omits some topicswhich we address.

K. Konishi and G. Pa⇤uti, Quantum Mechanics: A NewIntroduction, (OUP, 2009). This is a new text which includessome entertaining new topics within an old field.

L. D. Landau and L. M. Lifshitz, Quantum Mechanics:Non-Relativistic Theory, Volume 3, (Butterworth-Heinemann,3rd edition, 1981). Classic text which covers core topics at a levelthat reaches beyond the ambitions of this course.

F. Schwabl, Quantum Mechanics, (Springer, 4th edition, 2007).Best text for majority of course.













Syllabus •  Introduction to quantum mechanics •  History overview of quantum theory •  Review of classical mechanics •  Wave function and Schrödinger equation •  Postulates of quantum mechanics •  Time-independent Schrödinger equation •  One-dimensional time-independent problems •  Mathematical formalism •  Uncertainty principle •  Hydrogen atom •  Angular momentum •  Identical particles and quantum statistics

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Mathematical Requirements •  Prerequisites: PHYS 344 and PHYS 410

•  Linear Algebra: 1. complex number 2. vector, vector space 3. matrix, basic matrix operations 4. linear operators •  Calculus: derivative, integral

•  Differential equations: Linear differential equations See Appendix of your textbook if your math background needs to be refreshed and/or strengthened.

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Homework •  Developing problem-solving skills

–  There will be 13 homework assignments (20 points each). –  Final homework score will be calculated after dropping

the one with the lowest score. –  Problem sets will be assigned each Tuesday. –  The homework is due and has to be brought to the

lecture on Thursday of the following week. –  Students may discuss the problems with each other in a

general way but should not do the homework as a group effort. No carbon copy homework sets are acceptable. Further, the problem solutions should be clearly and neatly written on one side only of standard size paper. Your fellow students should be able to read, follow and understand the solutions. The quality of the presentation counts towards the grade.

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Exams and Grades •  Exams:

–  There will be one midterm exam and a final exam. All exams are closed-book.

–  Midterm Exam: October 17, 2013

•  Grades: –  The final grade will be determined on the following basis:

•  30% homework •  30% midterm exam •  40% final exam

–  We will use plus/minus letter grades. –  The exact cut-offs for letter grades will not be determined until the

end of the semester.

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Quantum Mechanics •  A. Einstein

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that He doesn't play dice.

•  N. Bohr If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.

•  R. Feynman I think I can safely say that nobody understands Quantum Mechanics.

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Introduction to Quantum Mechanics A law governing microscopic world

•  All objects are built of small common bricks

•  The behavior of large objects can be different from their elements

•  Classical physics describes the macroscopic world

•  Quantum physics describes the microscopic world

•  Classical physics can be considered as a natural limit of quantum mechanics by taking the Planck constant to be zero

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Quantum Mechanics •  The quantum mechanical world is VERY different!

–  Energy not continuous, but can take on only particular discrete values.

–  Light has particle-like properties, so that light can bounce off objects just like balls.

–  Particles also have wave-like properties, so that two particles can interfere just like light does.

–  Physics is not deterministic, but events occur with a probability determined by quantum mechanics.

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The Quantum Mechanics View •  All matter (particles) has wave-like properties

–  so-called particle-wave duality

•  Particle-waves are described in a probabilistic manner –  electron doesn‘t whiz around the nucleus, it has a probability

distribution describing where it might be found –  allows for seemingly impossible “quantum tunneling”

•  Some properties come in dual packages: can’t know both simultaneously to arbitrary precision –  called the Heisenberg Uncertainty Principle –  not simply a matter of measurement precision –  position/momentum and energy/time are example pairs

•  The act of “measurement” fundamentally alters the system –  called entanglement: information exchange alters a particle’s state

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History of Quantum Mechanics

Classical Physics •  Before 1900: Classical physics claimed a full victory

Newton’s Law and Gravity

Classical Electrodynamics

Classical Statistical Physics

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Atomic Hypothesis •  Around 1900: The atomic hypothesis became popular

•  1897: J.J. Thomson – discovery of the electron •  1905: E. Rutherford – atomic model

•  If the model is right, it is the end of classical mechanics.

• How can an atom be stable? • Energy would be lost by radiation. • Surrounding electrons would

have collapsed to nuclear. There must be new physical laws!

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•  1910: Millikan measures the electric charge – it’s quantized

Atomic Hypothesis •  Around 1900: The color of atoms

•  Different atoms glowed in different colors

•  Optical spectrum of atoms: characteristic of the elements

•  Balmer (1885) found an ordering principle in atomic spectra

Optical spectra of Calcium

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The Black Body Spectrum

•  Light radiated by an object characteristic of its temperature, not its surface color

•  Spectrum of radiation changes with temperature 1859 G. Kirchhoff

The Black Body Spectrum

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•  Measurements: O. Lummer, E. Pringsheim, H. Rubens, F. Kurlbaum

•  The wavelength of the peak of the blackbody distribution was found to follow

•  Wien’s displacement law: –  Peak wavelength shifts with temperature –  λmax is the wavelength at the curve’s peak –  T is the absolute temperature of the object

emitting the radiation

•  Stefan-Boltzmann law: Total energy radiated per unit surface area of a black body per unit time is proportional to fourth power of temperature.

λmax =

constantTemperature

Classical Theory

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Black-body radiation

In thermal equilibrium, radiation emitted by a cavity in frequencyrange ⌅ = c

� to ⌅ + d⌅ is proportional to mode density and fixed byequipartition theorem (kBT per mode):

Rayleigh-Jeans law ⌃(⌅, T ) d⌅ =8⇧⌅2

c3kBT d⌅

i.e. ⌃(⌅, T ) increases without bound – UV catastrophe.

e.g. emission from cosmicmicrowave background(T ⇧ 2.728K )

Experimentally, distribution conforms to Rayleigh-Jeans law at lowfrequencies but at high frequencies, there is a departure!

Emission from cosmic microwave background (T = 2.728 K)

Experimentally, distribution conforms to Rayleigh-Jeans law at low frequencies but at high frequencies, there is a departure!

Rayleigh-Jeans law:

In thermal equilibrium, radiation emitted by a cavity in frequency range ν = c/λ to ν + dν is proportional to mode density and fixed by equipartition theorem (kBT per mode):

Black-body radiation

In thermal equilibrium, radiation emitted by a cavity in frequencyrange ⌅ = c

� to ⌅ + d⌅ is proportional to mode density and fixed byequipartition theorem (kBT per mode):

Rayleigh-Jeans law ⌃(⌅, T ) d⌅ =8⇧⌅2

c3kBT d⌅

i.e. ⌃(⌅, T ) increases without bound – UV catastrophe.

e.g. emission from cosmicmicrowave background(T ⇧ 2.728K )

Experimentally, distribution conforms to Rayleigh-Jeans law at lowfrequencies but at high frequencies, there is a departure!

i.e. ρ(ν,T) increases without bound – UV catastrophe

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Classical Theory

•  Classical physics had absolutely no explanation for this

•  Only explanation they had gave ridiculous answer

•  Amount of light emitted became infinite at short wavelength –  Ultraviolet catastrophe

Explanation by Q.M. •  Blackbody radiation spectrum could only be

explained by quantum mechanics.

•  Radiation made up of individual photons, each with energy (Planck’s constant) x (frequency).

•  Very short wavelengths have very high energy photons.

•  Minimum energy is 1 photon.

•  For shorter wavelengths even 1 photon is too much energy, so shortest wavelengths have very little intensity.

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Heat Radiation

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•  Black body radiation: E=hν Each mode carries discrete energy quanta:

Planck: I can characterize the whole procedure as an act of desperation, since, by nature I am peaceable and opposed to doubtful adventures. However, I had already fought for six years (since 1894) with the problem of equilibrium between radiation and matter without arriving at any successful result. I was aware that this problem was of fundamental importance in physics, and I knew the formula describing the energy distribution . . . hence a theoretical interpretation had to be found at any price, however high it might be.

M. Planck

Black-body radiation: Planck’s resolution

Planck: for each mode, ⌅, energy is quantized in units of h⌅, whereh denotes the Planck constant. Energy of each mode, ⌅,

⌃ (⌅)⌥ =

⇧⇧n=0 n h⌅ e�nh⇥/kBT

⇧⇧n=0 e�nh⇥/kBT

=h⌅

eh⇥/kBT � 1

Leads to Planck distribution:

⌃(⌅, T ) =8⇧⌅2

c3⌃ (⌅)⌥ =

8⇧h⌅3

c3

1

eh⇥/kBT � 1

recovers Rayleigh-Jeans law as h ⌅ 0 and resolves UV catastrophe.

Parallel theory developed to explain low-temperature specific heat ofsolids by Debye and Einstein.

Einstein: Light energy is quantized Photoelectric effects (1905)

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•  Current is only generated with the frequency of the light higher than a threshold.

•  No matter how large is the power of light, there is no electric current if the frequency of the light is below the threshold.

Specific heat in solids (1907)

•  Specific heat is a constant in classical statistical physics •  Einstein-Debye model: lattice vibration energy is quantized •  Specific heat is temperature dependent.

Atomic Spectra

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•  Studies of electric discharge in low-pressure gases reveals that atoms emit light at discrete frequencies.

•  What caused spectra of atoms to contain discrete “lines” –  it was apparent that only a small set of

optical frequencies (wavelengths) could be emitted or absorbed by atoms

•  Each atom has a distinct “fingerprint” •  Light only comes off at very specific

wavelengths –  or frequencies –  or energies

•  Note that hydrogen (bottom), with only one electron and one proton, emits several wavelengths

Bohr Atom (1913)

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•  Electron orbits in an atom are quantized

•  Each orbits have their own energy

•  Angular momentum of an electron in one of these orbits is quantized in units of Planck’s constant.

•  Discrete values reflect emission of photons with energy En − Em = hν equal to difference between allowed electron orbits

•  The absorbed light is exactly such that the photon carries the energy difference between the two orbits Bohr-Sommerfeld quantization:

Atomic spectra: Bohr model

Studies of electric discharge inlow-pressure gases reveals that atomsemit light at discrete frequencies.

For hydrogen, wavelength followsBalmer series (1885),

⇤ = ⇤0

⇤1

4� 1

n2

⌅

Bohr (1913): discrete values reflect emission of photons with energyEn � Em = h⌅ equal to di⇤erence between allowed electron orbits,

En = �Ryn2

Angular momenta quantized in units of Planck’s constant, L = n�.

Atomic spectra: Bohr model

Studies of electric discharge inlow-pressure gases reveals that atomsemit light at discrete frequencies.

For hydrogen, wavelength followsBalmer series (1885),

⇤ = ⇤0

⇤1

4� 1

n2

⌅

Bohr (1913): discrete values reflect emission of photons with energyEn � Em = h⌅ equal to di⇤erence between allowed electron orbits,

En = �Ryn2

Angular momenta quantized in units of Planck’s constant, L = n�.

Limitation of Bohr’s Theory •  It is a theory with conjectures •  Lack of real calculation power •  Limit to hydrogen-type atoms •  Do not know how to extend it to more

complicated system

Anyway, Bohr’s theory fundamentally changes our view of world.

•  System is characterized by states. •  The physics is determined by final and initial states. •  Energy is quantized.

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The Birth of Quantum Mechanics •  Bohr’s theory failed:

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At the turn of the year from 1922 to 1923, the physicists looked forward with enormous enthusiasm towards detailed solutions of the outstanding problems, such as the helium problem and the problem of the anomalous Zeeman effects. However, within less than a year, the investigation of these problems revealed an almost complete failure of Bohr's atomic theory. (Quote from Jagdish Mehra and Helmut Rechenberg: monumental history of quantum mechanics)

•  De Broglie wave = particle (1923, age 31) •  Heisenberg Matrix theory (1925, age 23) •  Erwin Schrodinger wave equations (1926) •  Max Born: probabilistic interpretation of quantum mechanics (1926) •  P.A.M Dirac: relativistic quantum mechanics (1926, age 22) •  Linus Pauling: identical particle principle (1931, age 30)

•  Rapid development in 1920’s (modern quantum mechanics)

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Key Experiments •  Black-body radiation •  Photoelectric effect •  Compton scattering •  Atomic spectra •  Electron diffraction

Pre-quantum Problems •  Why was red light incapable of knocking electrons out of certain

materials, no matter how bright –  yet blue light could readily do so even at modest intensities –  called the photoelectric effect –  1905 Einstein explained in terms of photons, and won Nobel Prize

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Energy of Light •  Quantization also applies to other physical systems

–  In the classical picture of light (EM wave), we change the brightness by changing the power (energy/sec).

–  This is the amplitude of the electric and magnetic fields. –  Classically, these can be changed by arbitrarily small

amounts

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Quantization of Light

•  Possible energies for green light (λ=500 nm)

E=hf

E=2hf

E=3hf

E=4hf

–  One quantum of energy: one photon

–  Two quanta of energy two photons

–  etc

•  Think about light as a particle rather than wave.

Quantum mechanically, brightness can only be changed in steps, with energy differences of hf.

The Particle Perspective

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•  Light comes in particles called photons. •  Energy of one photon is E=hf

f = frequency of light

•  Photon is a particle, but moves at speed of light! –  This is possible because it has zero mass

•  Zero mass, but it does have momentum: –  Photon momentum p=E/c

One quantum of green light

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One quantum of energy for 500 nm light

E = hf = hc

λ=

6.634 × 10−34 J − s( ) × 3 × 108 m / s( )500 × 10−9 m

= 4 × 10−19 J

Quite a small energy! Quantum mechanics uses new ‘convenience unit’ for energy:

1 electron-volt = 1 eV = |charge on electron| x (1 volt) = (1.602x10-19 C) x (1 volt) 1 eV = 1.602x10-19 J

In these units, E(1 photon green) = (4x10-19 J) x (1 eV / 1.602x10-19 J) = 2.5 eV

E = hc

λ= constant [in eV − nm]

wavelength [in nm]= 1240 eV − nm

500 nm= 2.5 eV

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Photon properties of light •  Photon of frequency f has energy hf

•  Red light made of ONLY red photons

•  The intensity of the beam can be increased by increasing the number of photons/second

•  Photons/second = energy/second = power

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But light is a wave! •  Light has wavelength, frequency, speed

–  Related by fλ = speed

•  Light shows interference phenomena –  Constructive and destructive interference

L

Shorter path

Longer path

Light beam

Foil with two narrow slits

Recording plate

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Wave behavior of light: interference

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Particle behavior of light: Photoelectric Effect •  A metal is a bucket holding electrons •  Electrons need some energy in order to jump out of the

bucket.

A metal is a bucket of electrons

Energy transferred from the light to the electrons.

Electron uses some of the energy to break out of bucket.

Remainder appears as energy of motion (kinetic energy).

Light can supply this energy.

Unusual experimental results

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•  Not all kinds of light work •  Red light does not eject electrons

More red light doesn‘t either

No matter how intense the red light, no electrons ever leave the metal

Until the light wavelength passes a certain threshold, no electrons are ejected.

Wavelength dependence

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Long wavelength: NO electrons ejected

Short wavelength: electrons ejected

Hi-energy photons Lo-energy photons

Threshold depends on material

Einstein’s explanation

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•  Einstein said that light is made up of photons, individual ‘particles’, each with energy hf.

•  One photon collides with one electron - knocks it out of metal.

•  If photon doesn’t have enough energy, cannot knock electron out.

•  Intensity ( = # photons / sec) doesn’t change this.

Minimum frequency (maximum wavelength) required to eject electron

Photon properties of light •  Photon of frequency f has energy hf •  Red light made of ONLY red photons •  The intensity of the beam can be increased by

increasing the number of photons/second •  Photons/second = energy/second = power

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•  Interaction with matter •  Photons interact with matter one at a time. •  Energy transferred from photon to matter. •  Maximum energy absorbed is photon energy.

Photoelectric effect •  When metal exposed to EM radiation,

above a certain threshold frequency, light is absorbed and electrons emitted.

•  von Lenard (1902) observed that energy of electrons increased with light frequency (as opposed to intensity).

•  Einstein (1905) proposed that light composed of discrete quanta (photons): K.E.max = hν − W

•  Einstein’s hypothesis famously confirmed by Millikan in 1916

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Photoelectric e�ect

When metal exposed to EM radiation, above acertain threshold frequency, light is absorbed andelectrons emitted.

von Lenard (1902) observed that energy ofelectrons increased with light frequency (asopposed to intensity).

Einstein (1905) proposed that light composed ofdiscrete quanta (photons): k.e.max = h⌅ �W

Einstein’s hypothesis famouslyconfirmed by Millikan in 1916

Photoelectric e�ect

When metal exposed to EM radiation, above acertain threshold frequency, light is absorbed andelectrons emitted.

von Lenard (1902) observed that energy ofelectrons increased with light frequency (asopposed to intensity).

Einstein (1905) proposed that light composed ofdiscrete quanta (photons): k.e.max = h⌅ �W

Einstein’s hypothesis famouslyconfirmed by Millikan in 1916

PHOTOELECTRIC EFFECT

When UV light is shone on a metal plate in a vacuum, it emits charged particles (Hertz 1887), which were later shown to be electrons by J.J. Thomson (1899).

Electric field E of light exerts force F=-eE on electrons. As intensity of light increases, force increases, so KE of ejected electrons should increase.

Electrons should be emitted whatever the frequency ν of the light, so long as E is sufficiently large.

For very low intensities, expect a time lag between light exposure and emission, while electrons absorb enough energy to escape from material.

Classical expectations

H. Hertz J.J. Thomson

I

Vacuum chamber

Metal plate

Collecting plate

Ammeter

Potentiostat

Light, frequency ν

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PHOTOELECTRIC EFFECT (cont)

The maximum KE of an emitted electron is then

maxK h Wν= −Work function: minimum energy needed for electron to escape from metal (depends on material, but usually 2-5eV)

Planck constant: universal constant of nature

346.63 10 Jsh −= ×

Einstein

Millikan

Verified in detail through subsequent experiments by Millikan

Maximum KE of ejected electrons is independent of intensity, but dependent on ν

For ν<ν0 (i.e. for frequencies below a cut-off frequency) no electrons are emitted

There is no time lag. However, rate of ejection of electrons depends on light intensity.

Actual results:

E hν=

Einstein’s interpretation (1905):

Light comes in packets of energy (photons)

An electron absorbs a single photon to leave the material

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SUMMARY OF PHOTON PROPERTIES

E = hν

p = h

λ= hν

c

E = ω p = k = h

2π k = 2π

λ

Energy and frequency

•  Also have relation between momentum and wavelength

E2 = p2c2 + m2c4

c = λν

•  Relation between particle and wave properties of light

•  Relativistic formula relating energy and momentum

E = pc For light and

Also commonly write these as

ω = 2πνangular frequency

wavevector h bar

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Photon Energy •  Light is quantized into packets called photons •  Photons have associated:

–  frequency, ν –  speed, c (always) –  energy: E = hν

•  higher frequency photons ν è higher energy –  momentum: p = hν/c

•  The constant, h, is Planck’s constant –  has tiny value of: h = 6.63 ×10-34 J·s

•  Every particle or system of particles can be defined in quantum mechanical terms –  and therefore have wave-like properties

•  The quantum wavelength of an object is: ν = h/p (p is momentum)

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Summary of Photoelectric Effect •  Explained by quantized light.

•  Red light is low frequency, low energy.

•  (Ultra)violet is high frequency, high energy.

•  Red light will not eject electron from metal, no matter how intense. –  Single photon energy hf is too low.

•  Need ultraviolet light

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•  Evidence for wave-particle duality •  Photoelectric effect •  Compton effect

•  Electron diffraction •  Interference of matter-waves

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Wave Particle Duality

Neither Wave nor Particle •  Light in some cases shows properties

typical of waves

•  In other cases shows properties we associate with particles

•  Conclusion: –  Light is not a wave, or a particle, but something we

haven’t thought about before.

–  Reminds us in some ways of waves.

–  In some ways of particles.

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Do an interference experiment again.

But turn down the intensity until only ONE photon at a time is between slits and screen

Photon interference?

?

Only one photon present here

Is there still interference?

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Single-photon interference

1/30 sec exposure

1 sec exposure

100 sec exposure

Probabilities

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•  We detect absorption of a photon at camera.

•  Cannot predict where on camera photon will arrive.

•  Position of an individual photon hits is determined probabilistically.

•  Photon has a probability amplitude through space. Square of this quantity gives probability that photon will hit particular position on detector.

•  The photon is a probability wave! The wave describes what the particle does.

Compton Scattering

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•  Photons can transfer energy to beam of electrons.

•  Determined by conservation of momentum, energy.

•  Compton awarded 1927 Nobel prize for showing that this occurs just as two balls colliding.

Arthur Compton 1936

Compton Scattering

X-ray source

Target

Crystal (selects wavelength)

Collimator (selects angle)

θ

Compton (1923) measured intensity of scattered X-rays from solid target, as function of wavelength for different angles. He won the 1927 Nobel prize.

Result: peak in scattered radiation shifts to longer wavelength than source. Amount depends on θ (but not on the target material). A.H. Compton, Phys. Rev. 22 409 (1923)

Detector

Compton

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Compton’s explanation: “billiard ball” collisions between particles of light (X-ray photons) and electrons in the material

Classical picture: oscillating electromagnetic field causes oscillations in positions of charged particles, which re-radiate in all directions at same frequency and wavelength as incident radiation.

Change in wavelength of scattered light is completely unexpected classically

θ

pe

p ′νBefore After

Electron

Incoming photon

pν

scattered photon

scattered electron

Oscillating electron Incident light wave Emitted light wave

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Compton Scattering

Conservation of energy Conservation of momentum

hν + mec

2 = h ′ν + pe2c2 + me

2c4( )1/2

pν =

hλ

i = p ′ν + pe

′λ − λ = hmec

1− cosθ( )= λc 1− cosθ( ) ≥ 0

λc = Compton wavelength = h

mec= 2.4×10−12 m

From this Compton derived the change in wavelength

θ

pe

p ′νBefore After

Electron

Incoming photon

pν

scattered photon

scattered electron

Compton Scattering

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Compton scattering

In 1923, Compton studied scattering ofX-rays from carbon target.

Two peaks observed: first at wavelengthof incident beam; second varied withangle.

If photons carry momentum,

p =h⌅

c=

h

⇤

electron can recoil and be ejected.

Energy/momentum conservation:

�⇤ = ⇤⌅ � ⇤ =h

mec(1� cos ⇥)

Note that, at all angles there is also an un-shifted peak.

This comes from a collision between the X-ray photon and the nucleus of the atom

′λ − λ = h

mN c1− cosθ( ) 0

mN mesince

Compton Scattering

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In 1924 Einstein wrote: “There are therefore now two theories of light, both indispensable, and … without any logical connection.”

•  Evidence for wave-nature of light •  Diffraction and interference •  Evidence for particle-nature of light •  Photoelectric effect •  Compton effect

•  Light exhibits diffraction and interference phenomena that are only explicable in terms of wave properties

•  Light is always detected as packets (photons); if we look, we never observe half a photon

•  Number of photons proportional to energy density (i.e. to square of electromagnetic field strength)

WAVE-PARTICLE DUALITY OF LIGHT

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Matter Waves

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•  If light waves have particle-like properties, maybe matter has wave properties?

•  de Broglie postulated that the wavelength of matter is related to momentum as

•  This is called the de Broglie wavelength.

λ =hp

Louis de Broglie, Nobel prize 1929

We have seen that light comes in discrete units (photons) with particle properties (energy and momentum) that are related to the wave-like properties of frequency and wavelength.

λ = h

p

In 1923 Prince Louis de Broglie postulated that ordinary matter can have wave-like properties, with the wavelength λ related to momentum p in the same way as for light

de Broglie wavelength

de Broglie relation

h = 6.63×10−34 Js Planck’s constant

Prediction: We should see diffraction and interference of matter waves

De Broglie

NB: wavelength depends on momentum, not on the physical size of the particle

Matter Waves

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•  Classical waves have certain modes with fixed boundary

•  From Plank-Einstein: Light with frequency ω

•  Consider a particle as wave with wavelength defined as

De Broglie’s Particle-wave

E = ωp = ω / c = h / λ

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ph /=λ

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Wavelengths of massive objects

De Broglie wavelength =

€

λ =hp

p=mv for a nonrelativistic (v<<c) particle with mass.

€

λ =hmv

•  We argue that applies to everything

•  Photons and footballs both follow the same relation.

•  Everything has both wave-like and particle-like properties

λ = h

p

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This is very small

•  1 nm = 10-9 m •  Wavelength of red light = 700 nm •  Spacing between atoms in solid ~ 0.25 nm •  Wavelength of football = 10-26 nm

•  What makes football wavelength so small?

λ = h

p= h

mvLarge mass, large momentum short wavelength

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Wavelength of electron

•  Need less massive object to show wave effects •  Electron is a very light particle •  Mass of electron = 9.1x10-31 kg

λ = h

p= h

mv= 6 × 10−34 J − s

9 × 10−31 kg( ) × velocity( )•  Wavelength depends on mass and velocity

•  Larger velocity, shorter wavelength

65

Wavelength of 1 eV electron •  Fundamental relation is wavelength =

•  Need to find momentum in terms of kinetic energy. •  p = mv, so

λ = h

p

Ekinetic =

p2

2m p = 2mEkinetic

λ = h

p= h

2mEkinetic

= hc2mc2Ekinetic

•  Wavelength of electron with 50eV kinetic energy

K = p2

2me

= h2

2meλ2 ⇒λ = h

2meK= 1.7 ×10−10 m

•  Wavelength of Nitrogen molecule at room temperature

K = 3kT2

, Mass = 28mu

λ = h3MkT

= 2.8×10−11m

•  Wavelength of Rubidium(87) atom at 50nK

λ = h

3MkT= 1.2×10−6 m

De Broglie Wave Length

66

Davisson-Germer Experiment

67

•  Diffraction of electrons from a nickel single crystal

•  Established that electrons are waves

54 eV electrons (λ=0.17nm)

Bright spot: constructive interference

Davisson: Nobel Prize 1937

68

Wave reflection from crystal

•  If electron are waves they can interfere •  Interference of waves reflecting from different atomic

layers in the crystal. •  Difference in path length ~ spacing between atoms

side view

Reflection from top plane Reflection

from next plane

69

Particle interference •  Used this interference idea to learn

about the structure of matter

•  100 eV electrons: λ = 0.12nm –  Crystals also the atom

•  10 GeV electrons: –  Inside the nucleus, 3.2 fermi, 10-6 nm

•  10 GeV protons: –  Inside the protons and neutrons: 0.29 fermi

λ = 1240 eV − nm

2 × m0 MeV1KE

Electron Diffraction

70

C. Davisson L. Germer

Davisson, C. J., "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928)

The Davisson-Germer experiment: scattering a beam of electrons from a Ni crystal. Davisson got the 1937 Nobel prize.

At fixed accelerating voltage (fixed electron energy) find a pattern of sharp reflected beams from the crystal

At fixed angle, find sharp peaks in intensity as a function of electron energy

G.P. Thomson performed similar interference experiments with thin-film samples

θi

θi

The Davisson-Germer experiment (1927)

Interpretation: Similar to Bragg scattering of X-rays from crystals

a

θi acosθ i

Electron Diffraction (cont)

71

acosθr

Path difference:

Constructive interference when

Note difference from usual “Bragg’s Law” geometry: the identical scattering planes are oriented perpendicular to the surface Note θi and θr not

necessarily equal

Electron scattering dominated by surface layers

a(cosθr − cosθ i )

a(cosθr − cosθ i ) = nλ

θr

The Double Slit Experiment

particle? wave?

72

The Double Slit Experiment

73

d sinθ

Originally performed by Young (1801) to demonstrate the wave-nature of light. Has now been done with electrons, neutrons, He atoms among others.

D

θ d

Detecting screen

Incoming coherent beam of particles (or light)

y

Alternative method of detection: scan a detector across the plane and record number of arrivals at each point

For particles we expect two peaks, for waves an interference pattern

Experimental Results

74

Neutrons, A. Zeilinger et al. 1988 Reviews of Modern Physics 60 1067-1073

He atoms: O. Carnal and J. Mlynek 1991 Physical Review Letters 66 2689-2692

C60 molecules: M. Arndt et al. 1999 Nature 401 680-682 With multiple-slit grating

Without grating

Interference patterns can not be explained classically - clear demonstration of matter waves

Fringe visibility decreases as molecules are heated. L. Hackermüller et al. 2004 Nature 427 711-714

75

Atoms and Quanta: Bohr’s Theory

Planetary model of atom

76

•  Positive charge is concentrated in the center of the atom (nucleus)

•  Atom has zero net charge: –  Positive charge in nucleus cancels

negative electron charges.

•  Electrons orbit the nucleus like planets orbit the sun

•  (Attractive) Coulomb force plays role of gravity

nucleus

electrons

Planetary Model and Radiation •  Circular motion of orbiting electrons

causes them to emit electromagnetic radiation with frequency equal to orbital frequency.

•  Same mechanism by which radio waves are emitted by electrons in a radio transmitting antenna.

•  In an atom, the emitted electromagnetic wave carries away energy from the electron. –  Electron predicted to continually lose energy. –  The electron would eventually spiral into the nucleus;

will take ~10-10 s for an orbit of size 10-10 m –  However most atoms are stable!

77

78

Atoms and Photons •  Experimentally, atoms do emit electromagnetic

radiation, but not just any radiation! •  In fact, each atom has its own ‘fingerprint’ of

different light frequencies that it emits.

Hydrogen Emission Spectrum

79

•  Hydrogen is simplest atom –  One electron orbiting around one

proton.

•  The Balmer Series of emission lines empirically given by

n = 3, λ = 656.3 nm n = 4, λ = 486.1 nm

n=3 n=4

1λm

= RH

122 − 1

n2

⎛⎝⎜

⎞⎠⎟ Balmer (1885)

Balmer’s Formula (1885)

80

,...4,3,5810.109677

)121(/1

)4

(

1

22

2

2

==

−==

−=

− ncmRn

R

Gnn

H

Hλν

λ

How good is the formula? Example: 15233.21(exp) vs 15233.00 (th) for n=3 20564.77(exp) vs 20564.55 (th) for n=4

RH: Rydberg constant

More Spectra

,...1'...,2,1'5810.109677

)1'1(/1

)'

(

1

22

22

2

+===

−==

−=

−

nnncmRnn

R

Gnn

n

H

Hλν

λ

81

n’, n: principal quantum numbers

1906 Lyman 1908 Paschen 1922 Brackett

The Bohr Hydrogen Atom

82

•  Retained ‘planetary’ picture: one electron orbits around one proton

•  Only certain orbits are stable

•  Radiation emitted only when electron jumps from one stable orbit to another.

•  Here, the emitted photon has an energy of hν = Einitial - Efinal

Stable orbit #2

Stable orbit #1

Einitial

Efinal Photon

Hydrogen Emission •  This says hydrogen emits only

photons of a particular wavelength, frequency

•  Photon energy E = hf, so this means a particular energy.

•  Conservation of energy: –  Energy carried away by photon is lost by the orbiting

electron.

83

84

Energy Levels •  Instead of drawing orbits, we can just indicate the energy

an electron would have if it were in that orbit. Zero energy

n=1

n=2

n=3 n=4

€

E1 = −13.612 eV

€

E2 = −13.622 eV

€

E3 = −13.632 eV

Ene

rgy

axis

Energy quantized!

Emitting and Absorbing Light

85

Zero energy

n=1

n=2

n=3 n=4

€

E1 = −13.612 eV

€

E2 = −13.622 eV

€

E3 = −13.632 eV

n=1

n=2

n=3 n=4

€

E1 = −13.612 eV

€

E2 = −13.622 eV

€

E3 = −13.632 eV

Absorbing a photon of correct energy makes electron jump to higher quantum state.

Photon absorbed hf=E2-E1

Photon emitted hf=E2-E1

Photon is emitted when electron drops from one quantum state to another

Energy Conservation for Bohr Atom •  Each orbit has a specific energy

En=-13.6/n2 •  Photon emitted when electron

jumps from high energy to low energy orbit.

Ei – Ef = h f

•  Photon absorption induces electron jump from low to high energy orbit.

Ef – Ei = h f •  Agrees with experiment!

86

Bohr’s Solution

3/123/2

0

3/4

0

22 )(

)4(21

421 ω

πεπεω me

remrE −=−=

87

12

mv2 = 12

nhf

mvr = n•  Classical orbits:

•  Energy of electron:

•  Bohr’s bold postulates: •  The classical motion of electron in atom is still valid. However, only

discrete orbits with certain energy En is allowed. •  The motion of the electrons in these quantized orbits is radiationless. •  The light is emitted or absorbed when the electron transfers from one

orbit to the other. With increasing orbital radius r, the law should become identical to classical physics -- Bohr’s Correspondence Principle

mv2

r= mrω 2 = e2

4πε 0

1r 2

En = −Rhc / n2

En − En' = hω

Bohr’s Solution

88

1320

4

3/1233/2

0

3/42

31

318.109737)8/(

))/2(()4(2

/

/2

−

−

==

−=−=

=Ω=→−

cmchmeR

nRcmenRhcE

hhnRhcEE

n

nn

ε

πε

ω

nrmlhnRhc

===

2

3/2ω

ω

•  Classical: light frequency = classical electron orbiting frequency

•  From Bohr correspondence principle, we have

•  Quantization of angular momentum:

nlprpnhnr

==== /2 λπ

Standing waves

89

Electron Waves in an Atom

•  Electron is a wave.

•  In the orbital picture, its propagation direction is around the circumference of the orbit.

•  Wavelength = h / p (p=momentum, and energy determined by momentum)

•  How can we think about waves on a circle?

Hydrogen Atom Waves •  These are the five lowest energy

orbits for the one electron in the hydrogen atom.

•  Each orbit is labeled by the quantum number n.

•  The radius of each is na0.

•  Hydrogen has one electron: the electron must be in one of these orbits.

•  The smallest orbit has the lowest energy. The energy is larger for larger orbits.

90

91

Quantized Energy Levels

•  Quantized momentum

•  Energy = kinetic

•  Or Quantized Energy E = p2

2m=

npo( )2

2m= n2Eo

�

En = n2Eo

p = h

λ= n

h2L

= npo

Ene

rgy

n=1 n=2 n=3 n=4

n=5

Hydrogen Atom Energies

92

•  Wavelength gets longer in higher n states and the kinetic energy goes down (electron moving slower)

•  Potential energy goes up more quickly, also:

Zero energy

n=1

n=2

n=3 n=4

�

E1 = −13.612 eV

�

E2 = −13.622 eV

�

E3 = −13.632 eV

Ene

rgy

�

En = −13.6n2 eV

λ = h

p= hc

2 m0 Ekinetic

Ekinetic =

(hc)2

2m0λ2

λ ∝ n

Epot ∝

1r 2 ∝ 1

n2

Beyond Bohr’s Theory

hce

knnz

nRhcE kn

0

2

2

22

2,

2137/1

...))4/3/(1[

εα

α

==

+−+−=

93

•  Sommerfeld’s extention of the Bohr Model (Relativistic mass change) •  Semiclassical quantization rule:

∫ = npdq

The Hydrogen Atom •  When the mathematical machinery of quantum mechanics is

turned to the hydrogen atom, the solutions yield energy levels in exact agreement with the optical spectrum –  Emergent picture is one of probability distributions describing where

electrons can be

•  Probability distributions are static –  electron is not thought to whiz around atom: it’s in a “stationary

state” of probability

•  Separate functions describe the radial and angular pattern –  http://hyperphysics.phy-astr.gsu.edu/hbase/hydwf.html

The energy levels of hydrogen match the observed spectra, and fall out of the mathematics of quantum mechanics

94

De Broglie’s interpretation of Bohr’s orbit

95

2πr = nλ = nh / ppr = l = n

96

•  Every particle has a wavelength

•  However, particles are at approximately one position. –  Works if the particles has a superposition nearby of wavelengths

rather than one definite wavelength

•  Heisenberg uncertainty principle –  However particle is still spread out over small volume in

addition to being spread out over several wavelengths

Particle and wave

�

λ = hp

• 440 Hz + 439 Hz + 438 Hz + 437 Hz + 436 Hz

�

Δx( ) Δp( ) ~ /2

The Test of Quantum Mechanics •  Bohr-Einstein Debate •  Einstein-Podolsky-Rosen (EPR) The measurement of a particle at one location could reveal

instantly information about a second particle far away. •  Bell inequality (1964): tested. Quantum mechanics

holds.

97

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PHYSICS 460 Quantum Mechanics