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: – Jun Cheng – Office: PHYS 285 – Phone: 765 464-9735 – Email: [email protected] – Office hours: Wednesday: 11:00-12:00 am, Thursday: 9:00-10:00 am, 3:00-4:00 pm

Purdue University, Physics 460 1 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 Purdue University, Physics 460 2 Syllabus

• Introduction to quantum mechanics • History overview of quantum theory • Wave function and Schrödinger Equation • Postulates of Quantum Mechanics • Time-independent Schrödinger Equation • One-dimensional time-independent problems • Mathematical formalism • Heisenberg quantum mechanics and uncertainty principle • Hydrogen atom • Angular momentum • Identical particles and quantum statistics

Purdue University, Physics 460 3 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 460 4 Homework

• Developing problem-solving skills – There will be approximately 12 homework assignments. – 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.

Purdue University, Physics 460 5 Exams and Grades • Exams: – There will be one midterm exam and a final exam. All exams are closed-book. – Midterm Exam: October 20, 2011

• 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.

Purdue University, Physics 460 6 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.

Purdue University, Physics 460 7 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

Purdue University, Physics 460 8 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 460 9 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 460 10 History of Quantum Mechanics

Purdue University, Physics 460 11 Classical Physics • Before 1900: Classical physics claimed a full victory

Classical Electrodynamics

Classical Statistical Newton’s Law and Gravity Physics

Purdue University, Physics 460 12 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 460 13 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 460 14 The Black Body Spectrum

• Light radiated by an object characteristic of its temperature, not its surface color. • Spectrum of radiation changes with temperature

Purdue University, Physics 460 15 The Black Body Spectrum

• The wavelength of the peak of the blackbody distribution was found to follow constant !max = Temperature • Peak wavelength shifts with temperature

• λ max is the wavelength at the curve’s peak • T is the absolute temperature of the object emitting the radiation

Purdue University, Physics 460 16 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 460 17 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 const) 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 460 18 Heat Radiation Planck • Black body radiation: Each mode carries discrete energy quanta: E=hν

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.

Purdue University, Physics 460 19 Einstein: Light energy is quantized Photoelectric effects: (1905) Specific heat in solids (1907)

• Current is only generated with the frequency of • Specific heat is a constant in classical statistical physics the light higher than a threshold. • Einstein-Debye model: lattice vibration energy is quantized • Specific heat is temperature dependent. • No matter how large is the power of light, there is no electric current if the frequency of the light is below the threshold.

Purdue University, Physics 460 20 Bohr atom (1913)

• Electron orbits in an atom are quantized

• Each orbits have their own energy

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

Purdue University, Physics 460 21 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 (1925) • 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 460 22 The victory of the weird theory

• Without Quantum Mechanics, we could never have designed and built: – semiconductor devices • computers, cell phones, etc. – lasers • CD/DVD players, bar-code scanners, surgical applications – MRI (magnetic resonance imaging) technology – nuclear reactors – atomic clocks (e.g., GPS navigation) • Physicists didn’t embrace quantum mechanics because it was gnarly, novel, or weird – it’s simply that the #$!&@ thing worked so well

Purdue University, Physics 460 23 Key Experiments

Purdue University, Physics 460 24 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 460 25 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 460 26 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 460 27 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 460 28 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 ! 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 ! wavelength [in nm] 500 nm Purdue University, Physics 460 29 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

Purdue University, Physics 460 30 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

Light beam Longer path

Foil with two Recording narrow slits plate

Purdue University, Physics 460 31 Wave behavior of light: interference

Purdue University, Physics 460 32 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.

Light can supply this energy. Energy transferred from the light to the electrons. Electron uses some of the energy to break out of bucket. A metal is a bucket of electrons. Remainder appears as energy of motion (kinetic energy).

Purdue University, Physics 460 33 Unusual experimental results

• 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.

Purdue University, Physics 460 34 Wavelength dependence

Short wavelength: electrons Long wavelength: ejected NO electrons ejected

Threshold depends on material

Hi-energy photons Lo-energy photons

Purdue University, Physics 460 35 Einstein’s explanation

• 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

Purdue University, Physics 460 36 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 460 37 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 460 38 Pre-quantum problems, cont.

• 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 460 39 Wave Particle Duality

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

• Electron diffraction • Interference of matter-waves

Purdue University, Physics 460 40 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.

Purdue University, Physics 460 41 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 460 42 Single-photon interference

1/30 sec 1 sec 100 sec exposure exposure exposure

Purdue University, Physics 460 43 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 460 44 Hertz J.J. Thomson 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).

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 460 45 PHOTOELECTRIC EFFECT (cont) Einstein Actual results: Einstein’s interpretation (1905): Maximum KE of ejected electrons is independent of intensity, but Light comes in packets dependent on ν of energy (photons) Millikan For ν<ν0 (i.e. for frequencies below a cut-off frequency) no Eh= ν 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 =ν − 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 460 46 SUMMARY OF PHOTON PROPERTIES

• Relation between particle and wave properties of light Energy and frequency E = h! • Also have relation between momentum and wavelength • Relativistic formula relating E 2 = p2c2 + m2c4 energy and momentum For light E = pc and c = !" h h" p = = ! c Also commonly write these as wavevector h bar angular frequency 2! h E = !! p = !k ! = 2"# k = ! = " 2! Purdue University, Physics 460 47 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 460 48 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 460 49 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 460 50 COMPTON SCATTERING (cont)

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 p"! scattered photon Incoming photon θ p! Electron p scattered electron e Purdue University, Physics 460 51 COMPTON SCATTERING (cont) Before After p "! scattered photon Incoming photon θ p! Electron p e scattered electron Conservation of energy Conservation of momentum 1/2 h h! + m c2 = h!" + p2c2 + m2c4 p ˆi p p e ( e e ) ! = = !# + e " From this Compton derived the change in wavelength h "! # " = (1# cos$ ) mec 1 cos 0 = "c ( # $ ) % h ! = Compton wavelength = = 2.4 "10#12 m c m c e Purdue University, Physics 460 52 COMPTON SCATTERING (cont)

Note that, at all angles there is also an unshifted peak.

This comes from a collision between the X-ray photon and the nucleus of the atom h "! # " = 1# cos$ ! 0 m c ( ) N since m ! m N e

Purdue University, Physics 460 53 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 460 54 Matter Waves

• 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

h " = p

• This is called the de Broglie wavelength. ! Nobel prize, 1929 Purdue University, Physics 460 55 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 ! = "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 460 56 De Broglie’s Particle-wave

• Classical waves have certain modes with fixed boundary

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

λ = h / p

Purdue University, Physics 460 57 Wavelengths of massive objects h De Broglie wavelength = " = p h " = p=mv for a nonrelativistic mv (v<

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

Purdue University, Physics 460 58 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? h h ! = = Large mass, large momentum p mv short wavelength

Purdue University, Physics 460 59 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 ! = = = p mv 9 10#31 kg velocity ( " ) " ( ) • Wavelength depends on mass and velocity • Larger velocity, shorter wavelength

Purdue University, Physics 460 60 Wavelength of 1 eV electron h • Fundamental relation is wavelength = ! = p • Need to find momentum in terms of kinetic energy. • p = mv, so p2 E = p = 2mEkinetic kinetic 2m h h hc ! = = = p 2mE 2mc2 E kinetic kinetic Purdue University, Physics 460 61 De Broglie Wave Length • Wavelength of electron with 50eV kinetic energy

p2 h2 h K = = " ! = = 1.7 #10$10 m 2m 2m ! 2 2m K e e e • Wavelength of Nitrogen molecule at room temperature

3kT K = , Mass = 28m 2 u h ! = = 2.8 "10#11m 3MkT • Wavelength of Rubidium(87) atom at 50nK h ! = = 1.2 "10#6 m 3MkT Purdue University, Physics 460 62 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 460 63 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 460 64 Particle interference • Used this interference idea to to learn about the structure of matter 1240 eV " nm 1 ! = 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 460 65 Electron Diffraction The Davisson-Germer experiment (1927)

The Davisson-Germer experiment: Davisson G.P. Thomson θi 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 460 66 Electron Diffraction (cont)

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

θi Path difference: acos! i a(cos cos ) !r " !i

θr Constructive interference when a a(cos cos ) n !r " !i = # acos !r Electron scattering dominated by surface layers Note difference from usual “Bragg’s Law” geometry: the identical scattering planes are

Note θi and θr not oriented perpendicular to the surface necessarily equal Purdue University, Physics 460 67 The Double Slit Experiment

particle? wave?

Purdue University, Physics 460 68 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 sin! across the plane d θ and record number of Incoming coherent arrivals at each beam of particles point (or light) Detecting screen D For particles we expect two peaks, for waves an interference pattern

Purdue University, Physics 460 69 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

Fringe C60 molecules: M. Arndt et al. 1999 visibility Nature 401 decreases as 680-682 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 460 70 Atoms and Quanta: Bohr’s Theory

Purdue University, Physics 460 71 Planetary model of atom

• Positive charge is concentrated in the center of the atom electrons (nucleus) • Atom has zero net charge: – Positive charge in nucleus cancels nucleus negative electron charges. • Electrons orbit the nucleus like planets orbit the sun • (Attractive) Coulomb force plays role of gravity

Purdue University, Physics 460 72 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!

Purdue University, Physics 460 73 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.

Purdue University, Physics 460 74 Optical Spectra of Hydrogen Atom

There must be new physical laws! Discrete numbers: 656.3, 486.1, 434(nm),… Purdue University, Physics 460 75 Hydrogen Emission Spectrum

• Hydrogen is simplest atom – One electron orbiting around one n=4 n=3 proton.

• The Balmer Series of emission lines empirically given by 1 # 1 1 & R = H % 2 " 2 ( !m $ 2 n ' Balmer (1885) n = 4, λ = 486.1 nm n = 3, λ = 656.3 nm

Purdue University, Physics 460 76 Balmer’s Formula (1885)

n2 λ = ( )G n2 − 4 1 1 ν =1/ λ = R ( − ) H 22 n2 −1 RH =109677.5810cm ,n = 3,4,...

RH: Rydberg constant 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

Purdue University, Physics 460 77 More Spectra

n2 λ = ( 2 2 )G n − n' 1906 Lyman 1 1 ν =1/ λ = R ( − ) 1908 Paschen H n'2 n2 −1 1922 Brackett RH =109677.5810cm n'=1,2...,n = n'+1,... n’, n: principal quantum numbers

Purdue University, Physics 460 78 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 460 79 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.

Purdue University, Physics 460 80 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=4 13.6 E = " eV n=3 3 32

13.6 E eV n=2 2 = " 2 ! 2

!

13.6 E = " eV Energyaxis n=1 1 12 Energy quantized! Purdue University, Physics 460 81 ! Emitting and Absorbing Light Zero energy

n=4 n=4 13.6 13.6 E eV E = " eV n=3 3 = " 2 n=3 3 32 3

13.6 13.6 E = " eV E 2 = " 2 eV n=2 2 2 n=2 2 ! 2 !

Photon ! emitted ! Photon absorbed hf=E2-E1 hf=E2-E1 13.6 13.6 E1 = " eV E1 = " eV n=1 2 n=1 12 1

Photon is emitted when electron Absorbing a photon of correct drops from one quantum! state to energy makes electron! jump to another higher quantum state.

Purdue University, Physics 460 82 Energy Conservation for Bohr Atom

• Each orbit has a specific energy 2 En=-13.6/n • 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!

Purdue University, Physics 460 83 Bohr’s Solution

1 2 1 mv2 e2 1 mv = nhf • Classical orbits: = mr! 2 = 2 2 r 4 r 2 "# 0 mvr = n!

2 4/3 • Energy of electron: 1 2 e 1 e 2 1/3 E = mrω − = − 2/3 (mω ) 2 4πε 0 r 2(4πε 0 ) 2 En = ! Rhc / n • Bohr’s bold postulates: E E h n ! n' = " • 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

Purdue University, Physics 460 84 Bohr’s Solution

• Classical: light frequency = classical electron orbiting frequency 2πr = nλ = nh / p Standing waves pr = l = n • From Bohr correspondence principle, we have

3 En − En−1 → 2Rhc / n = hΩ = hω 4/3 2 e 3 2 1/3 En = −Rhc / n = − 2/3 (m(2Rc / n ) ) 2(4πε 0 ) 4 2 3 −1 R = me /(8ε 0 h c) =109737.318cm • Quantization of angular momentum:

2Rhc / n3 = hω l = mω 2r = n

Purdue University, Physics 460 85 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?

Purdue University, Physics 460 86 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.

Purdue University, Physics 460 87 Quantized Energy Levels

• Quantized momentum h h p = = n = np ! 2L o • Energy = kinetic 2 2 n=5 p (npo ) 2 E = = = n E 2m 2m o n=4 • Or Quantized Energy Energy n=3 2 E n = n E o n=2 n=1

Purdue University, Physics 460 88

Hydrogen Atom Energies h hc ! = = Zero energy p 2 m E 0 kinetic n=4 2 13.6 E eV (hc) n=3 3 = ! 2 E = ! " n 3 kinetic 2m ! 2 0 13.6 E = ! eV • Wavelength gets longer in n=2 2 2 2 higher n states and the kinetic energy goes down (electron

moving slower) Energy • Potential energy goes up 13.6 more quickly, also: E = ! eV n=1 1 12 1 1 13.6 E ! ! E = ! eV pot r 2 n2 n n 2 Purdue University, Physics 460 89

Beyond Bohr’s Theory

• Sommerfeld’s extension of the Bohr Model (Relativistic mass change)

− Rhc α 2 z 2 E = [1+ (n / k − 3/ 4) +...) n,k n2 n2 e2 α =1/137 = 2 hc ε 0 • Semi-classical quantization rule:

∫ pdq = n

Purdue University, Physics 460 90 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 The energy levels of hydrogen match the observed spectra, and fall out of the mathematics of quantum mechanics

• 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 Purdue University, Physics 460 91 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 460 92 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.

Purdue University, Physics 460 93