Atomic01 - University of Cape Town

PHY322S Atomic Physics 050716.1200
UCT PHY322S
Atomic Physics 01
July 2005
Professor David Aschman
Room 512, Physics Department
University of Cape Town
mailto:dga@science.uct.ac.za
file:/atomic/test.pdf
http://www.phy.uct.ac.za/courses/phy322s/atomic.htm .
1

PHY322S Atomic Physics .
Philosophy of lecturing and teaching
Pre-lecture reading assigned (see website)
Concepts and problems to think about and discuss
Questions in class
Lecture
Reading
discussion with class tutor and lecturer
Problem sets: attempt them early in the week
2

PHY322S Atomic Physics .
Atomic Physics: Websites
http://www.phy.uct.ac.za/courses/phy322s/
Others: see PHY322S website, or Google
3

PHY322S Atomic Physics .
Quantum Mechanics: Textbooks
Cassels JM, Basic Quantum Mechanics (Macmillan)
Griffiths D, Intro to Quantum Mechanics (Prentice Hall)
Rae A, Quantum Mechanics (IoP)
Feynman R, Lectures on Physics Vol3 (Addison Wesley)
Greiner W, Quantum Mechanics (Springer Verlag)
Bransden B & Joachain C, Intro to Quantum Mechanics (Longmans)
Eisberg R & Resnick R, Quantum Physics of Atoms . . . (Wiley)
Brehm and Mullin, Introduction to the Structure of Matter (Wiley)
4

PHY322S Atomic Physics .
Atomic Physics: Textbooks
Foot C, Atomic Physics (Oxford)
Haken H & Wolf H, Physics of Atoms and Quanta (Springer)
Branden B & Joachain C, Physics of Atoms and Molecules (Longmans)
Woodgate G, Elementary Atomic Structure (Oxford)
5

PHY322S Atomic Physics
Atoms
link http://www.sljus.lu.se/stm/NonTech.html
Silicon atoms on
surface seen by
atomic force
microscope
(Lund)
6

PHY322S Atomic Physics .
Atomism
Greeks - four elements: earth water air fire.
A’tom: uncuttable. Lucretius, . . . , Dalton
Spectral lines - Balmer, . . .
Thomson - electron
Radioactivity - Bequerel, Rutherford
Atom → electrons and nucleus (Bohr, Rutherford)
Quantum structure: Balmer, Bohr, Rutherford, Heisenberg, Schrodinger,
Pauli, Dirac
7

PHY322S Atomic Physics .
Nucleus - protons and neutrons. Bohr, Wheeler, Wigner
. . . Complicated force. Liquid drop model. Nuclear shell model.
Deformation, . . . A rich mesoscopic quantum fluid. The strong
internucleon force - potential or QCD?
Proton, neutron → quark model
Quarks → (super)strings? → ??
Leptons: electron, muon, neutrinos . . .
Radiation: photons, . . . [gauge bosons]
Dark matter. Dark energy. [Quintessence ??]
8

PHY322S Atomic Physics .
Problems with atom as classical object
All hydrogen atoms alike: r = a 0 ∼ 0.53 × 10 −10 m.
Classical atom (Coulomb force provides centripetal force):
mv 2 /r = (1/4πɛ 0 )e 2 /r 2
Problems:
1) Atom could be any size: If r(t) is solution; scale by k and k 2 r(t/k 3 ) is
ALSO a solution. Who sets k?
2) Atom is unstable. Accelerate charge radiates, so spirals in. Lifetime
∼ 10 −11 s
/...
9

PHY322S Atomic Physics .
Atom as a quantum object
Atom is a QUANTUM object. Enter . Bohr quantizes angular
momentum mva 0 = giving
a 0 = 2
me 2 = 0.53 × 10−10 m
Energy is −13.6 eV. Binding energy. Is atom big or small? How strong
is electromagnetic interaction? Is motion non-relativistic? We must
interpret a 0 .
10

PHY322S Atomic Physics .
Atomic size from uncertainty principle
Assume electron confined to atomic box of size a. Uncertainty principle
says momentum p ∼ /a, so non-relativistic kinetic energy is p 2 /2m.
Add Coulomb energy to get total
E ∼
2
2ma − e 2
2 (4πɛ 0 )a
Choose a to get a 0 that minimizes E, ie set dE/da = 0
a 0 = 4πɛ 0 2
e 2 m
( = 4πɛ0 c
)(
)
e 2 mc
= α −1 λ e
where fine structure constant α = (1/4πɛ 0 )(e 2 /c) ∼ 1/137
and electron Compton wavelength λ e = /mc ∼ 3.85 × 10 −13 m
Atomic size a 0 = (137)(3.85 × 10 −13 ) ≃ 0.53 × 10 −10 m.
11

PHY322S Atomic Physics .
Physical quantities with m, , c
For a particle of mass m, we can use physical constants and c, to
form dimensionally correct quantities
m = [mass]
mc = [momentum]
mc 2 = [energy]
mc 2 / = [time] −1
/mc 2 = [time]
/mc = [length]
12

PHY322S Atomic Physics .
Natural units: the factor c
To evaluate an expression in natural units where [ = c = 1], put in
factors and c to get known groupings, eg mc 2 , then finally replace c
factors with
c = (2π) −1 (6.6 × 10 −34 )(1.6 × 10 −19 ) −1 (3 × 10 8 )(10 9 ) eV nm
= 197 eV nm
= 197 MeV fm
13

PHY322S Atomic Physics .
Tentative outline of the course
Early quantum ideas
Blackbody radiation and Planck. Photoelectric effect. Einstein equation. de Broglie
relation, electron diffraction. Compton scattering.
Steps towards wave mechanics
Wave-particle duality, Uncertainty principle. Schrödinger equation (time dependent,
independent). Intepretation of wavefunction.
One-dimensional time-independent problems
Square well potential (infinite, finite). Probability flux. Potential barrier and step.
Reflection and transmission. Tunnelling. Wavepackets. Simple harmonic oscillator.
14

PHY322S Atomic Physics .
The formal basis of quantum mechanics
Postulates of quantum mechanics operators, observables, eigenvalues, eigenfunctions.
Hermitian operators and the Expansion Postulate.
Angular momentum in quantum mechanics
Operators, eigenvalues and eigenfunctions of ˆl 2 and ˆl z .
The hydrogen atom
Separation of space and time parts of the 3D Schrödinger equation for a central field.
Radial wave equation and series method solution. Degeneracy and spectroscopic
notation.
/. . .
15

PHY322S Atomic Physics .
Electron spin and total angular momentum
Magnetic moment of electron due to orbital motion. The Stern-Gerlach experiment.
Electron spin and complete set of quantum numbers for the hydrogen atom. Addition
of angular momentum quantum numbers. Total spin and orbital angular momentum
quantum numbers S , L, J.
Emission and absorption of radiation by atoms
Interaction of atoms with an EM field. Selection rules for radiative transitions in
hydrogen and complex atoms.
16

PHY322S Atomic Physics .
Blackbody radiation & ultraviolet catastrophe
(a) Spectrum of blackbody radiation & Rayleigh-Jeans law
(b) Spectrum as a function of temperature
17

PHY322S Atomic Physics .
Wave particle duality
Evidence for wave-particle duality:
Photoelectric effect
Compton effect
Electron diffraction
Interference of matter-waves
Consequence:
Heisenberg uncertainty principle
18

PHY322S Atomic Physics .
Photoelectric effect
UV light shining on metal plate in a vacuum, causes emission of
charged particles (Hertz 1887); later shown to be electrons by J.J.
Thomson (1899).
Hertz J.J. Thomson
Classical expectations: 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 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.
19

PHY322S Atomic Physics .
Photoelectric effect: Einstein’s hypothesis
Experiment: Maximum KE of ejected electrons is independent of
intensity, but dependent on ν. For ν < ν 0 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.
Einstein interpretation (1905): Light comes in packets of energy
(photons) E = hν, An electron absorbs a single photon to leave the
material
The maximum kinetic energy of an emitted electron is E k = hν − W
Planck constant h = 6.63 × 10 −34 Js. Work function W ∼ 3 eV.
Verified in detail through subsequent experiments by Millikan.
20

PHY322S Atomic Physics .
Relation between particle and wave properties of light
Energy and frequency E = hν
Relativistic formula relating energy and momentum E 2 = (pc) 2 + (mc 2 ) 2
Momentum related to wavelength p = h/λ
For light: E = pc and c = λν and p = h/λ = hν/c
Common: = h/2π E = ω ω = 2πν k = 2π/λ
21

PHY322S Atomic Physics .
Compton scattering
Compton, Phys. Rev. 22,409 (1923) scattered X-rays from solid target;
intensity as function of wavelength for different angles. (1927 Nobel).
Shift to longer wavelength of peak in scattered radiation depends on θ
(but not on the target material).
22

PHY322S Atomic Physics .
Compton scattering
Classically, expect no change in frequency of radiation.
Comptons explanation: billiard ball collisions between particles of light
(X-ray photons) and electrons in the material
Conservation of energy: hν + m e c 2 = hν ′ + (p 2 ec 2 + m 2 ec 4 ) 1/2
Conservation of momentum: p = (h/λ)î = p ν ′ + p e
23

PHY322S Atomic Physics .
whence . . .
λ ′ − λ =
h (1 − cos θ)
m e c
Compton wavelength of the electron λ c = (h/(m e c)) = 2.4 × 10 −12 m
At all angles an unshifted peak arises from collisions between X-ray
photon and the nucleus of the atom
λ ′ − λ =
h (1 − cos θ) ∼ 0
m N c
since m N ≫ m e
24

PHY322S Atomic Physics .
Wave particle duality
There are therefore now two theories of light, both indispensable, and
without any logical connection. - Einstein (1924)
Evidence for wave-nature of light: Diffraction and interference
Evidence for particle-nature of light:
effect
Photoelectric effect, Compton
25

PHY322S Atomic Physics .
Matter waves
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.
In 1923 Prince Louis de Broglie postulated that ordinary matter can
have wave-like properties, with the wavelength related to momentum
in the same way as for light
λ = h p
Prediction: we should see diffraction and interference of matter waves.
de Broglie wavelength depends on momentum, not on the physical size
of the particle
26

PHY322S Atomic Physics .
Estimation of some de Broglie wavelengths
Wavelength of electron with 50 eV kinetic energy
K = p2
2m e
=
h2
2m e λ 2 → λ =
h
√ 2me K = 1.7 × 10−10 m
Wavelength of nitrogen molecule at room temperature
K = (3/2)kT, M = 28m u λ = h/ √ 3MkT = 2.8 × 10 −11 m
Wavelength of rubidium-87 atom at 50nK
λ = h/ √ 3MkT = 1.6 × 10 −6 m
27

PHY322S Atomic Physics .
Electron diffraction: Davisson-Germer experiment
Davisson-Germer (1927) scatter beam of electrons from a Ni crystal.
[Davison, ”Are Electrons Waves?,” Franklin Inst. J. 205, 597 (1928)]
At fixed accelerating voltage a pattern of sharp reflected beams from
the crystal. At fixed angle, find sharp peaks in intensity as a function
of electron energy (1937 Nobel). GP Thomson performed similar
interference experiments with thin-film samples.
28

PHY322S Atomic Physics .
Electron diffraction (cont.)
Interpretation: similar to Bragg
scattering of X-rays from crystals.
Path difference: Constructive
interference path difference a whole
number of wavelengths
a(cos θ r − cos θ i ) = nλ
Note difference from usual Bragg
Law geometry: scattering planes
are oriented perpendicular to
surface.
Note θ i not necessarily equal to θ r .
Electron scattering dominated by
surface layers.
29

PHY322S Atomic Physics .
Young’s double slit experiment
Young (1801) demonstrates the wave-nature of light. Has now been
done with electrons, neutrons, He atoms . . . . Expect two peaks for
particles, interference pattern for waves.
30

PHY322S Atomic Physics .
Double Slit - experiments
Interference patterns can not be explained classically. Clear
demonstration of matter wave hypothesis.
31

PHY322S Atomic Physics
Double slit experiment with helium atoms
Carnal & Mlynek, Phys. Rev. Lett.,66,(1991) p2689
Carnal & Mlynek, Phys. Rev. Lett., 66, (1991) p2689.
Experiment: He atoms at 83 K, with d = 8 µm and D = 64 cm.
Path difference d sin θ. Constructive interference: d sin θ = nλ.
Expected separation between maxima: ∆y = λD/d
Measured separation: ∆y = 8.2 µm
Energy K = 3kT/2, M = 4m u
Predicted de Broglie wavelength: λ = h/ √ 3MkT = 1.03 × 10 −10 m
Predicted separation: ∆y = 8.4 ± 0.8 µm
Good agreement with experiment.
32

PHY322S Atomic Physics .
Fringe spacing in double slit experiment
Maxima when d sin θ = nλ
D ≫ d so use small angle
approximation
θ ≃ nλ/d
so ∆θ ≃ λ/d
Position on screen: y = D tan θ ≃
Dθ
Separation between adjacent
maxima:
∆y ≃ D∆θ
∆y ≃ λD/d
33

PHY322S Atomic Physics
Double Slit experiment: interpretation
Feynman Lectures Vol 3 Ch2
Interference pattern persists at low intensity, even with one atom
arriving at a time:
Particle goes through both slits.
Wave behaviour shown by a single atom:
matter-wave inteferes with itself.
If we try to detect which slit particle went though, interference pattern
vanishes:
we cannot see particle and wave nature at the same time.
The importance of the two-slit experiment has been memorably
summarized by Richard Feynman: . . . phenomenon which is
impossible, absolutely impossible, to explain in any classical way, and
which has in it the heart of quantum mechanics. In reality it contains
the only mystery.
34

PHY322S Atomic Physics .
Double slit experiment - bibliography
See Physics World(September 2002) article http://physicsweb.
org where the double-slit experiment was voted the most beautiful
experiment in physics.
35

PHY322S Atomic Physics .
Heisenberg γ-ray microscope
The microscope is an imaginary device to measure the position y and
momentum p of a particle. Resolving power of lens: ∆y ≥ λ/θ
36

PHY322S Atomic Physics .
Heisenberg microscope (cont.)
Photons transfer momentum to the particle from which they scatter.
Magnitude of p is same before and after collision.
Uncertainty of photon y-momentum equal to uncertainty in particle y-
momentum
−p sin(θ/2) ≤ p y ≤ p sin(θ/2)
Small angle approximation: ∆p y = 2p sin(θ/2) ≃ pθ
de Brogile gives p = h/λ so ∆p y ≃ hθ/λ
and ∆y ≥ λ/θ hence
∆p y ∆y ≃ h
This is the Heisenberg uncertainty principle.
37

PHY322S Atomic Physics .
Measurement and uncertainty
The thought experiment seems to imply that, while prior to experiment
we have well defined values, it is the act of measurement which
introduces the uncertainty by disturbing the particles position and
momentum.
It is widely accepted now that quantum uncertainty (lack of
determinism) is intrinsic to the theory.
38

PHY322S Atomic Physics .
Heisenberg uncertainty principle
Formally it is shown that
∆x∆p x ≥ /2∆y∆p y ≥ /2∆z∆p z ≥ /2
We cannot have simultaneous knowledge of conjugate variables such
as position and momenta.
Note, however, ∆x∆p y ≥ 0, so arbitary precision is possible in principle
for position in one direction and momentum in another etc.
39

PHY322S Atomic Physics .
Heisenberg uncertainty principle
There is also an energy-time uncertainty relation
∆E∆t ≥ /2
Transitions between energy levels of atoms are not perfectly sharp
in frequency. An atomic lifetime is typically τ ∼ 10 − 8 s. There is a
corresponding spread in the emitted photon energy or frequency.
40

PHY322S Atomic Physics .
Conclusions
Light and matter exhibit wave-particle duality.
Relation between wave and particle properties given by the de Broglie
relations
E = hν p = h/λ
Evidence for particle properties of light:
Photoelectric effect, Compton scattering
Evidence for wave properties of matter:
Electron diffraction; interference of matter waves
Heisenberg uncertainty principle limits simultaneous knowledge of
conjugate variables
∆x∆p x ≥ /2∆y∆p y ≥ /2∆z∆p z ≥ /2
41