302 2009 Lecture summaries and lecture notes.
These are the lecture notes for Physics 302 - Advanced
Mechanics taught from Townsend - A Modern Approach to Quantum Mechanics
How to read these lecture
Each page has 4 panels, representing a chunk of blackboard about 3 feet
wide. The sequence of panels on a page is TL, TR, BL, BR. A 55
minute lecture is usually ten
panels, 2.5 pages.
Note on errata:
be grateful, if you find a mistake, if you let me know so I can
note it here.
Kay Warner and John Daise for scanning these notes.
1 Why we need quantum mechanics. Physics of dipoles.
2 The g-factor. The Stern - Gerlach experiment. The ﬁrst three
3 Experiment 4. Interference. The quantum state vector.
4 Other basis vectors. Expectation values and uncertainty.
5 Freedom of representation and change of basis. Rotation
6 Unitary and Hermitian operators. Matrix representation of
7 Properties of Hermitian operators. Interpretation of
eigenvalues and eigenvectors of hermitian operators. Action of
rotations on vectors in lab and in Hilbert space.
8 Expectation values. Photon polarization.
9 Correspondence between photon polarization states and spin 1/2.
Circularly polarized light, birefringence and quarter wave plates.
10 Completeness of angular momentum operators for spin 1/2.
Classical and quantum penny flip - importance of non-commuting
11 Rotations do not commute. Commutation relations for angular
momentum. Physical meaning of commutation.
12 Total A.M. commutes with any component. Commutators and the
uncertainty principle. The Schwarz inequality.
13 Examples of the uncertainty relations for spin 1/2. Finding the
eigenvalues of angular momentum using commutators. Ladder operators.
14 Matrix elements of the angular momentum operators. Example:
Spin-1. The Kochen-Specker paradox and contextuality.
15 Dynamics. The time evolution operator and the Schroedinger
equation. The Hamiltonian: the generator of time translations is the
energy operator. Energy eigenstates time evolve by phase multiplication.
16 Time dependence of Expectation values. Hamiltonian for
spin 1/2 in a B field. Larmor precession.
17 Predicting observations in a Larmor precession experiment.
Measurement of g factor. High precision muon decay g-factor experiment.
Neutron Mach-Zender interferometry and rotation of spin half particles
18 Cancelled due to snow storm
19 Dynamics of spin 1/2 in transverse field. Rabi Oscillations.
Driven transverse fields and resonance. NMR.
20 Spin echo experiments.
21 Review. More particles - the direct or tensor product of states.
22 Hyperfine structure in H. Direct or tensor product of operators.
Matrix elements of Hamiltonian. Eigenstates of Hyperfine interaction
and the 21 cm line.
23 Addition of angular momentum. Angular momentum in the energy
eigenstates of the hyperfine interaction. Change of basis from tensor
product basis to total AM basis - the Clebsch-Gordan coefficients.
24 Addition of angular momentum. Rules for combination of
eigenvalues - selection rules and the Clebsch-Gordan coefficients.
Example: singlet states. The EPR experiment, Bertlemanns socks and
local reality. Spin
25 Quantum technology: Quantum key distribution, no cloning,
teleportation and Grover search.
26 Wave mechanics in 1-D. The position basis and the translation
operator. The generator of translations is momentum. Commutation
relations for position and momentum operators.
27 Momentum eigenstates. The momentum operator and its eigenstates
in the position basis. Relevant time and length scales for electrons
with a few eV of energy.
Lecture 28 Switching between momentum and position eigenstates -
the Fourier transform. General solution of Schroedinger equation
for free particle. Phase velocity and group velocity.
29 Dispersion relation for quantum particles. Wave packets. Group
velocity is packet velocity. Gaussian packets and dispersion time.
Classical objects do not disperse. Here is a mathematica demo of
an electron with 1eV of energy localized in a gaussian packet of width
10 nm. Another notebook
of the same thing.
30 Gaussian packets - stationary and moving. The Gaussian packet in
momentum space. Scattering with packets and with momentum eigenstates.
Boundary conditions for the Schroedinger equation and properties of
solution. Boundary matching at a potential step. Movie
of reflection. Movie
of resonant transmission. Movie
of Transmission. Movie of
tunneling through narrow barrier.
31 Energy eigenstates for the simple Harmonic oscillator. Ladder
operators and the number operator. The energy eigenvalues. Zero point
energy and the uncertainty principle. The energy eigenstates in the
position basis. Mathematica demo: packets in the simple harmonic
oscillator. Mathematica demo
of eigenstates, superpositions of energy eigenstates and classical
notebook of exact packet solution for the SHO. Mathematica demo of squeezed states.
32 Parity in the 1-D simple harmonic oscillator. Beyond 1-D:
translations and their generators. Example systems: 3D S.H.O., 3D
infinite spherical well, Electrons on Helium, Hydrogen Atom. The 3D
simple harmonic oscillator. Separation of variables. Degeneracy.
33 Spherical symmetry and orbital angular momentum. Rotations in
terms of translations - L is r cross p. Good quantum numbers for
spherically symmetric problems. Position basis representation of
orbital angular momentum operators. Phi dependence of Lz eigenfunctions.
34 Phi dependence of Lz eigenfunctions. Single valuedness and
constraint that m be integer. Other components of angular
momentum: L^2 and the scalar laplacian. Separation of variables for
spherically symmetric problems. The angular problem: ladder operators. Mathematica
35 Solving the angular problem using ladder operators. Probability
interpretation of angular eigenfunctions and normalization. The
spherical harmonics: properties and visualization. Mathematica demo:
spherical harmonics. The radial problem - behavior of radial
wavefunction at origin and centrifugal barrier.
36 Large r behavior of radial wavefunction. Series solution of
differential equations. The full solution: the laguerre polynomials.
Energy levels of the hydrogen atom. degeneracy in l and m. The Aufbau
principle, Pauli exclusion and the periodic table. Spectroscopy of H:
Lyman and Balmer series.
37 Degeneracies and other physics in the H atom. Relativistic
corrections to the kinetic energy. Non-degenerate perturbation theory.
First order correction to the energy. Example: first relativistic
correction to the SHO ground state.
38 First order correction to the wavefunction. Perturbation theory
at second order. Vanishing of first order perturbations due to
symmetry. Example: S.H.O. in an electric field. Degenerate perturbation
theory. Example: perturbation of ferromagnetically coupled spins
by a transverse coupling.
39 The Stark effect in Hydrogenic atoms. The second order shift to
the non-degenerate ground state. The first order shift to the
degenerate first excited state. Electrons on Helium: spectrum,