PHYSICS 360 Quantum Mechanics

Prof. Norbert Neumeister Department of Physics and Astronomy Purdue University

Spring 2020

http://www.physics.purdue.edu/phys360 Course Format • Lectures: – Time: Monday, Wednesday 9:00 – 10:15 – Lecture Room: PHYS 331 – Instructor: Prof. N. Neumeister – Office hours: Tuesday 2:00 – 3:00 PM (or by appointment) – Office: PHYS 372 – Phone: 49-45198 – Email: [email protected] (please use subject: PHYS 360) • Grader: – Name: Guangjie Li – Office: PHYS 6A – Phone: 571-315-3392 – Email: [email protected] – Office hours: Monday: 1:30 pm – 3:30 pm, Friday: 1:00 pm – 3:00 pm

Purdue University, Physics 360 1 Textbook

The textbook is: Introduction to Quantum Mechanics, David J. Griffiths and Darrell F. Schroeter, 3rd 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 Purdue University, Physics 360 2 AA fewA (randomfew recommended but but but recommended) recommended) recommended) books books booksBooks B. H. Bransden and C. J. Joachain, Quantum Mechanics, (2nd B. H. H.B. H. Bransden Bransden Bransden and andand C. C. C. J. J. J. Joachain,Joachain Joachain,, QuantumQuantumQuantum Mechanics Mechanics Mechanics,,, (2nd(2 (2ndnd edition, Pearson, 2000). Classic text covers core elements of edition,edition, Pearson, Pearson, Pearson, 2000). 2000). 2000).ClassicClassic Classic text texttext covers covers core core core elements elements elements of ofadvanced of advanced quantum mechanics; strong on atomic physics. advancedquantum quantum quantum mechanics; mechanics; mechanics; strong on strongatomic strong physics. on on atomic atomic physics. physics. S.S. Gasiorowicz, Gasiorowicz, QuantumQuantumQuantum Physics Physics Physics,,, (2nd (2nd (2nd edn. edn. edn. Wiley Wiley Wiley 1996, 1996, 1996, 3rd 3rd 3rd S. Gasiorowicz, Quantum Physics, (3rd edition, Wiley, 2003). edition,edition, Wiley, Wiley, Wiley, 2003). 2003). 2003). ExcellentExcellentExcellent text text text covers covers covers material material material at at at Excellent text covers material at approximately right level; but published approximatelyapproximately right right right level; level; level; but but but published published published text text text omits omits omits some some some topics topics topics text omits some topics which we address. whichwhich we we we address. address. address. K.K. Konishi KonishiK. Konishi and and andand G. G. G. G. Pa Pa Pa Paffuti⇤⇤⇤uti,uti,uti,, QuantumQuantumQuantumQuantum Mechanics: Mechanics: Mechanics:Mechanics: A A AA New New NewNew IntroductionIntroductionIntroduction,,, (OUP, (OUP, (OUP,, (Oxford 2009). 2009). 2009). UniversityThisThisThis is is is a a a Press, new new new text text text2009). which which which This includes includes is includes a new text somesomewhich entertaining entertaining entertaining includes some new new new entertaining topics topics topics within within within new an an antopics old old old field.within field. field. an old field. L.L. D. D. Landau Landau Landau and and and L. L. L. M. M. M. Lifshitz, Lifshitz, Lifshitz,QuantumQuantumQuantum Mechanics: Mechanics: Mechanics: L. D. Landau and L. M. Lifshitz, Quantum Mechanics: Non- Non-RelativisticNon-Relativistic Theory, Theory, Volume Volume 3 3,, (Butterworth-Heinemann, (Butterworth-Heinemann, Non-RelativisticRelativistic Theory Theory,, Volume Volume 3, 3(Butterworth, (Butterworth-Heinemann,-Heinemann, 3rd 3rd3rd edition, edition, 1981). 1981). ClassicClassic text text which which covers covers core core topics topics at at a a level level 3rdedition, edition, 1981). 1981). ClassicClassic text text which which covers covers core topics core at topics a level at that a level thatthat reaches reaches beyond beyond the the ambitions ambitions of of this this course. course. thatreaches reaches beyond beyond the theambitions ambitions of this of course. this course. F.F. Schwabl, Schwabl,F. SchwablQuantumQuantumQuantum, Quantum Mechanics Mechanics Mechanics Mechanics,,, (Springer, (Springer, (Springer,, (Springer, 4th 4th 4th4th edition, edition,edition, edition, 2007).2007). 2007). 2007). BestBestVery text text good for for for text majority majority majority for majority of of of course. course. course. of course. Purdue University, Physics 360 3 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

Purdue University, Physics 360 4 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.

Purdue University, Physics 360 5 Homework • Developing problem-solving skills – There will be 12 homework assignments (20 points each). – Final homework score will be calculated after dropping the two with the lowest score. – Problem sets will be assigned each Monday. – The homework is due and has to be brought to the lecture on Wednesday 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. Purdue University, Physics 360 6 Exams and Grades • Exams: – There will be two midterm exam and a final exam. All exams are closed-book. – Midterm Exams: Mar 2, 2020 and Apr 13, 2020 – Missing midterm: no makeups

• Grades: – The final grade will be determined on the following basis: • 30% homework • 30% midterm exams (15% each) • 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.

Purdue University, Physics 360 7 Quantum Theory

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

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

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. Purdue University, Physics 360 8 Introduction to Quantum Mechanics

A law governing the 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

Purdue University, Physics 360 9 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.

Purdue University, Physics 360 10 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

Purdue University, Physics 360 11 Introduction to Quantum Mechanics Essential ideas

1. Uncertainty principle: Conjugates quantities of a particle (ex: position & momentum) can not be known simultaneously within a certain accuracy limit 2. Quantization: The measurement of a physical quantity in a confined system results in quanta (the measured values are discrete) 3. Wave-particle duality: All particles can be described as waves (travelling both in space and in time) The state of the particle is given by a wave function Ѱ(x,t) 4. Extrapolation to classical mechanics: The laws of classical Newtonian mechanics are the extrapolation of the laws of quantum mechanics for large systems with very large number of particles

Purdue University, Physics 360 12 Quantum Mechanics: The Secret What we observe is much less than what actually exists What is the ‘state’ of a system? • Classical mechanics: position and velocity. • Quantum mechanics: the wave function.

Position and velocity are what you can observe.

But until you measure them, they don’t exist. Only the wave function does.

The wave function tells you the probability of measuring different values of position or velocity.

Purdue University, Physics 360 13 The Wave Function All information about a system is provided by the system’s wave function.

�(x) Pr(x) x x

Interesting facts about the wave function:

1. The wave function can be positive, negative, or complex-valued. 2. The squared amplitude of the wave function at position x is equal to the probability of observing the particle at position x. 3. The wave function can change with time. 4. The existence of a wave function implies particle-wave duality.

Purdue University, Physics 360 14 The Wave Function The wave function tells us the probability, but it’s not equal to the probability. • For every possible

observable outcome, position the wave function has wave function (or other observable) a value. • Wave functions can be positive or negative. Different contributions to the wave function can therefore either reinforce, or cancel each other out. • Probability of observing an outcome = (wave function)2. More precisely: wave functions are complex numbers, y = a + ib, and the probability is given by |y|2 = a2 + b2.

Purdue University, Physics 360 15 The Measurement Pascual Jordan: “Observations not only disturb what has to be measured, they produce it….We compel [the electron] to assume a definite position…. We ourselves produce the results of measurements.” John Bell, 1964: It makes an observable difference whether the particle had a precise (though unknown) position prior to the measurement, or not.

Purdue University, Physics 360 16 History of Quantum Mechanics

Purdue University, Physics 360 17 Classical Physics • Before 1900: Classical physics claimed a full victory

Classical Electrodynamics

Classical Statistical Newton’s Law and Gravity Physics

Purdue University, Physics 360 18 Classical Mechanics

Isaac Newton (1642 – 1727)

Lagrange Hamilton ! ! (1736 – 1813) (1805 – 1865) Fma= Lagrangian: LTVº- 2 ! ! dr Hamilton’s variational principle: Fm= dt 2 t2 ddJ==ò Ldt 0 t1 Hamiltonian: HqpLTVº-å !ii =+ i ¶H = q!i Force (cause) à acceleration (consequence) Hamiltonian ¶p equations of i motion ¶H =-p!i ¶qi Purdue University, Physics 360 19 Atomic Hypothesis

• Around 1900: The atomic hypothesis became popular

• 1897: J.J. Thomson – discovery of the electron • 1905: E. Rutherford – atomic model • 1910: Millikan measures the electric charge – it’s quantized

• 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!

Purdue University, Physics 360 20 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

Purdue University, Physics 360 21 Classical Physics: Light

• 1621: W. Snell - refraction

• 1664: R. Hooke - interference: color in thin films • 1665: F. Grimaldi - diffraction

• 1677: C. Huygens - wave theory • Light is a wave moving in ‘ether’ analogy: water waves

• 1704: I. Newton - light is particles!

• 1801-1814: T. Young, Fresnel - it must be a wave!

Purdue University, Physics 360 22 Light Maxwell’s equations, 1864 • Light is electromagnetic wave ! ! ! r div(E) = Ñ × E = e ! ! ! 0 div(B) = Ñ × B = 0

! ! ! ¶B curl(E) = Ñ ´ E = - ¶t ! ! ! ! é ! ¶E ù curl(B) = Ñ ´ B = µ0 êJ + e0 ú ë ¶t û

• EM wave can have any amplitude

Purdue University, Physics 360 23 Light

Problem: black body radiation Rayleigh Planck

Classical physics cannot Wienn

explain black body U density, Energy radiation spectrum Frequency Max Planck, 1900 (Nobel 1918): Energy consists of quanta! E = hn photon h - Planck’s constant, h = 6.626×10-34 J.s n - frequency, Hz (or s-1)

Purdue University, Physics 360 24 Failure of Classical Physics

Problems: • Photoelectric effect (Einstein): photons • Line spectrum of atoms • Quantized electronic orbitals, quantized momenta

• Double-slit experiment results Electron diffraction!

Purdue University, Physics 360 25 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

Purdue University, Physics 360 26 The Black Body Spectrum

• Measurements: O. Lummer, E. Pringsheim, H. Rubens, F. Kurlbaum • The wavelength of the peak of the blackbody distribution was found to follow constant λmax = Temperature • 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. Purdue University, Physics 360 27 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

Purdue University, Physics 360 28 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.

Purdue University, Physics 360 29 Heat Radiation M. Planck • Black body radiation: Each mode carries discrete energy quanta: E=hν

Planck: I can characterize the whole Black-body radiation:procedure as Planck’s an act of desperation, resolution since, by nature I am peaceable and opposed to doubtful adventures. However, I had Planck: for eachalready mode, fought⌅, energy for six isyears quantized (since 1894) in units of h⌅, where with the problem of equilibrium between h denotes the Planckradiation constant. and matter Energy without of arriving each mode,at any ⌅, successful result. Inh was⇥/k awareT that this ⇧ nh⌅ e B h⌅ (⌅) problem= n=0 was of fundamental = importance in nh⇥/kBT h⇥/kBT ⌃ ⌥ physics, n⇧and=0 eI knew the formulae describing 1 the ⇧energy distribution . . . hence a Leads to Plancktheoretical distribution:⇧ interpretation had to be found at any price, however high it might be. 8⇧⌅2 8⇧h⌅3 1 ⌃(⌅, T )= (⌅) = c3 ⌃ ⌥ c3 eh⇥/kBT 1 Purdue University, Physics 360 30 recovers Rayleigh-Jeans law as h 0 and resolves UV catastrophe. ⌅

Parallel theory developed to explain low-temperature specific heat of solids by Debye and Einstein. Atomic Spectra • 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

Purdue University, Physics 360 31 Atomic spectra: Bohr model Bohr AtomAtomic spectra: (1913) Bohr model

Studies• Electron of electric orbits discharge in in an atom are Studies of electric discharge in low-pressurequantized gases reveals that atoms low-pressure gases reveals that atoms emit light at discrete frequencies. emit light at discrete frequencies. • Each orbits have their own energy For hydrogen, wavelength follows For hydrogen, wavelength follows Balmer• Angular series (1885), momentum of an electron in one of these orbits is quantized in Balmer series (1885), 1 1 ⇤ = ⇤ units of0 Planck’s4 n2 constant. 1 1 ⇤ ⌅ ⇤ = ⇤ • Discrete values reflect emission of 0 4 n2 Bohr (1913): discrete values reflect emission of photons with energy ⇤ ⌅ photons with energy En − Em = hν E E = h⌅ equal to di⇤erence between allowed electron orbits, equaln m to difference between allowed Bohr (1913): discrete values reflect emission of photons with energy electron orbits Ry E E = h⌅ equal to di⇤erence between allowed electron orbits, En = n m n2 Ry • AngularThe absorbed momenta quantized light inis units exactly of Planck’s such constant, that L = n. E = n n2 the photon carries the energy difference between the two orbits Angular momenta quantized in units of Planck’s constant, L = n. Bohr-Sommerfeld quantization:

Purdue University, Physics 360 32 The Bohr Hydrogen Atom

• Retained ‘planetary’ picture: one electron orbits around one proton E • Only certain orbits are stable initial • Radiation emitted only when Photon electron jumps from one Efinal stable orbit to another. • Here, the emitted photon has an energy of hν = Einitial - Efinal

Stable orbit #2 Stable orbit #1

Purdue University, Physics 360 33 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.

Purdue University, Physics 360 34 The Birth of Quantum Mechanics

• Bohr’s theory failed:

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)

• Rapid development in 1920’s (modern 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) Purdue University, Physics 360 35 Quantum Mechanics Early hints at the dawn of the 20th century

Max Planck: Albert blackbody Einstein: radiation photons

Henri Becquerel, Marie & Pierre Curie: radioactivity

Purdue University, Physics 360 36 Introduction to Quantum Mechanics

Early 20th century: Some revolutionary ideas from bright minds…

Werner Heisenberg Erwin Schrödinger Wolfgang Pauli (1901-1976) (1887-1961) (1900- 1958)

Uncertainty Principle Schrödinger Equation Pauli exclusion principle

Purdue University, Physics 360 37 Schrödinger Equation • Classical physics: dx2 ! ¶ V mF==- dt2 ¶ x • Quantum physics: 1926: Schrödinger equation ¶Y !22¶Y iV! = – +Y ¶t 2mx ¶ 2 Erwin Rudolf Josef (in one dimension) Alexander Schrödinger m – mass of the particle (1987 – 1961) V(x,t) – potential energy

! ºh 2p = 1.054572´ 10-34 J × s Purdue University, Physics 360 38 Key Experiments

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

Purdue University, Physics 360 39 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

Purdue University, Physics 360 40 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

Purdue University, Physics 360 41 Quantization of Light Quantum mechanically, brightness can only be changed in steps, with energy differences of hf. • Possible energies for green light (λ=500 nm)

– One quantum of energy: one photon – Two quanta of energy two photons E=4hf

– etc E=3hf

• Think about light as a particle E=2hf rather than wave. E=hf

Purdue University, Physics 360 42 The Particle Perspective

• 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

Purdue University, Physics 360 43 One Quantum of Green Light One quantum of energy for 500 nm light 6.634 × 10−34 J − s × 3 × 108 m / s hc ( ) ( ) −19 E = hf = = = 4 × 10 J λ 500 × 10−9 m 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 hc constant [in eV − nm] 1240 eV − nm E = = = = 2.5 eV λ wavelength [in nm] 500 nm Purdue University, Physics 360 44 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

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

Purdue University, Physics 360 45 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). H. Hertz J.J. Thomson Vacuum Light, frequency ν Classical expectations chamber Electric field E of light exerts force Collecting Metal F=-eE on electrons. As intensity of plate plate 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. I Ammeter For very low intensities, expect a time lag between light exposure and emission, Potentiostat while electrons absorb enough energy to escape from material.

Purdue University, Physics 360 46 Photoelectric Effect (cont)

Actual results: Einstein’s Einstein interpretation (1905): Maximum KE of ejected electrons is independent of intensity, but Light comes in packets dependent on ν of energy (photons)

For ν<ν0 (i.e. for frequencies below a cut-off frequency) no Eh= n Millikan electrons are emitted An electron absorbs a There is no time lag. However, single photon to leave rate of ejection of electrons the material depends on light intensity.

The maximum KE of an emitted electron is then

KhWmax =n - Work function: minimum Verified in detail Planck constant: energy needed for electron to through subsequent universal constant escape from metal (depends experiments by of nature on material, but usually 2-5eV) Millikan h =6.63´ 10-34 Js Purdue University, Physics 360 47 Summary of Photon Properties • Relation between particle and wave properties of light Energy and frequency E = hν • Also have relation between momentum and wavelength 2 2 2 2 4 • Relativistic formula relating E = p c + m c energy and momentum For light E = pc and c = λν h hν p = = λ c Also commonly write these as wavevector h bar angular frequency 2π h E = ω p = k ω = 2πν k =  = 2 Purdue University, Physics 360 λ π 48 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) Purdue University, Physics 360 49 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

Purdue University, Physics 360 50 Wave Particle Duality

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

• Electron diffraction • Interference of matter-waves

Purdue University, Physics 360 51 Photon interference?

Do an interference Only one photon present here experiment again. But turn down the intensity until only ONE photon at a time is between slits and screen ?

Is there still interference?

Purdue University, Physics 360 52 Single-Photon Interference

1/30 sec 1 sec 100 sec exposure exposure exposure

Purdue University, Physics 360 53 Probabilities

• 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.

Purdue University, Physics 360 54 Compton Scattering

• 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

Purdue University, Physics 360 55 Compton Compton Scattering Compton (1923) measured intensity of scattered X-rays from solid target, as function of wavelength for different angles. He won the 1927 Nobel prize.

X-ray source Collimator Crystal (selects angle) (selects wavelength)

θ Target

Result: peak in scattered radiation Detector shifts to longer wavelength than source. Amount depends on θ (but not on the target material). A.H. Compton, Phys. Rev. 22 409 (1923) Purdue University, Physics 360 56 Compton Scattering 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

Incident light wave Oscillating electron Emitted light wave

Compton’s explanation: “billiard ball” collisions between particles of light (X-ray photons) and electrons in the material Before After pν′ scattered photon Incoming photon θ pν Electron p scattered electron e Purdue University, Physics 360 57 Compton scattering

In 1923, Compton studied scattering of Compton ScatteringX-rays from carbon target. Before After p Two peaks observed: ν′ first at wavelength of incident beam;scattered second photon varied with Incoming photon angle. θ pν Electron If photons carry momentum, p scattered electron e h⌅ h Conservation of energy Conservationp = of= momentum c ⇤ 1/2 h hν + m c2 = hν′ + p2c2 + m2c4 p ˆi p p e ( e e ) electron canν = recoil= andν′ be+ ejected.e λ From this Compton derived the changeEnergy/momentum in wavelength conservation: h λ′ − λ = (1− cosθ ) h mec ⇤ = ⇤⌅ ⇤ = (1 cos ⇥) me c 1 cos 0 = λc ( − θ ) ≥ h λ = Compton wavelength = = 2.4 ×10−12 m c m c e Purdue University, Physics 360 58 Compton Scattering

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 h λ′ − λ = 1− cosθ  0 m c ( ) N since m  m N e

Purdue University, Physics 360 59 Wave-Particle Duality of Light

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) Purdue University, Physics 360 60 Matter Waves We have seen that light comes in discrete units (photons) with

particle properties (energy and momentum) that are related to the De Broglie wave-like properties of frequency and wavelength.

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 relation h Planck’s constant λ = −34 h = 6.63×10 Js de Broglie wavelength p

NB: wavelength depends on momentum, not on the physical size of the particle Prediction: We should see diffraction and interference of matter waves Purdue University, Physics 360 61 De Broglie’s Particle-Wave

• Classical waves have certain modes with fixed boundary

• From Plank-Einstein: Light with frequency ω E = ω p = ω / c = h / λ • Consider a particle as wave with wavelength defined as l = h / p

Purdue University, Physics 360 62 Wavelength of Electron

• Need less massive object to show wave effects • Electron is a very light particle • Mass of electron = 9.1x10-31 kg h h 6 × 10−34 J − s λ = = = p mv 9 10−31 kg velocity ( × ) × ( ) • Wavelength depends on mass and velocity • Larger velocity, shorter wavelength

Purdue University, Physics 360 63 Wavelength of 1 eV electron h • Fundamental relation is wavelength = λ = p • Need to find momentum in terms of kinetic energy. • p = mv, so p2 E = p = 2mEkinetic kinetic 2m h h hc λ = = = p 2mE 2mc2 E kinetic kinetic Purdue University, Physics 360 64 De Broglie Wave Length • Wavelength of electron with 50eV kinetic energy

p2 h2 h K = = ⇒ λ = = 1.7 ×10−10 m 2m 2m λ 2 2m K e e e • Wavelength of Nitrogen molecule at room temperature

3kT K = , Mass = 28m 2 u h λ = = 2.8 ×10−11m 3MkT • Wavelength of Rubidium(87) atom at 50nK h λ = = 1.2 ×10−6 m 3MkT Purdue University, Physics 360 65 Davisson-Germer Experiment • Diffraction of electrons Bright spot: constructive from a nickel single crystal interference • Established that electrons are waves Davisson Nobel Prize 1937

54 eV electrons (λ=0.17nm)

Purdue University, Physics 360 66 Wave Reflection from Crystal Reflection from Reflection top plane from next plane

side view • 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 Purdue University, Physics 360 67 Particle Interference • Used this interference idea to learn about the structure of matter 1240 eV − nm 1 λ = 2 m MeV KE × 0

• 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

Purdue University, Physics 360 68 Electron Diffraction The Davisson-Germer experiment (1927)

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

θi

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

Davisson, C. J., At fixed accelerating voltage (fixed "Are Electrons electron energy) find a pattern of sharp Waves?," Franklin reflected beams from the crystal Institute Journal 205, 597 (1928)

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

Purdue University, Physics 360 69 Electron Diffraction (cont)

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

θi Path difference: acosθ i a(cos cos ) θr − θi

θr Constructive interference when a a(cos cos ) n θr − θi = λ acos θr Electron scattering dominated by surface layers Note difference from usual “Bragg’s Law” geometry: the identical scattering planes Note θi and θr not are oriented perpendicular to the surface necessarily equal Purdue University, Physics 360 70 The Double Slit Experiment

particle? wave?

Purdue University, Physics 360 71 The Double Slit Experiment

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

Alternative method of y detection: scan a detector d sinθ across the plane d θ and record number of Incoming coherent arrivals at each beam of particles point Detecting (or light) screen D For particles we expect two peaks, for waves an interference pattern

Purdue University, Physics 360 72 Experimental Results 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. Fringe Arndt et al. 1999 visibility Nature 401 680- decreases as 682 molecules are With heated. L. multiple-slit Hackermüller grating et al. 2004 Nature 427 Without grating 711-714

Interference patterns can not be explained classically - clear demonstration of matter waves Purdue University, Physics 360 73