Physics 302 2009 Lecture summaries and lecture notes.

These are the lecture notes for Physics 302 - Advanced Quantum Mechanics taught from Townsend - A Modern Approach to Quantum Mechanics

How to read these lecture notes: 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: I'd be grateful, if you find a mistake, if you let me know so I can note it here.

Acknowledgements: Thanks to Kay Warner and John Daise for scanning these notes.

Lecture 1 Why we need quantum mechanics. Physics of dipoles.

Lecture 2 The g-factor. The Stern - Gerlach experiment. The ﬁrst three experiments.

Lecture 3 Experiment 4. Interference. The quantum state vector.

Lecture 4 Other basis vectors. Expectation values and uncertainty.

Lecture 5 Freedom of representation and change of basis. Rotation operators.

Lecture 6 Unitary and Hermitian operators. Matrix representation of operators.

Lecture 7 Properties of Hermitian operators. Interpretation of eigenvalues and eigenvectors of hermitian operators. Action of rotations on vectors in lab and in Hilbert space.

Lecture 8 Expectation values. Photon polarization.

Lecture 9 Correspondence between photon polarization states and spin 1/2. Circularly polarized light, birefringence and quarter wave plates.

Lecture 10 Completeness of angular momentum operators for spin 1/2. Classical and quantum penny flip - importance of non-commuting operations.

Lecture 11 Rotations do not commute. Commutation relations for angular momentum. Physical meaning of commutation.

Lecture 12 Total A.M. commutes with any component. Commutators and the uncertainty principle. The Schwarz inequality.

Lecture 13 Examples of the uncertainty relations for spin 1/2. Finding the eigenvalues of angular momentum using commutators. Ladder operators.

Lecture 14 Matrix elements of the angular momentum operators. Example: Spin-1. The Kochen-Specker paradox and contextuality.

Lecture 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.

Lecture 16 Time dependence of Expectation values. Hamiltonian for spin 1/2 in a B field. Larmor precession.

Lecture 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 `twice'.

Lecture 18 Cancelled due to snow storm

Lecture 19 Dynamics of spin 1/2 in transverse field. Rabi Oscillations. Driven transverse fields and resonance. NMR.

Lecture 20 Spin echo experiments.

Lecture 21 Review. More particles - the direct or tensor product of states.

Lecture 22 Hyperfine structure in H. Direct or tensor product of operators. Matrix elements of Hamiltonian. Eigenstates of Hyperfine interaction and the 21 cm line.

Lecture 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.

Lecture 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 operators notebook.

Lecture 25 Quantum technology: Quantum key distribution, no cloning, teleportation and Grover search.

Lecture 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.

Lecture 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.

Lecture 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.

Lecture 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.

Lecture 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 limit. Mathematica notebook of exact packet solution for the SHO. Mathematica demo of squeezed states.

Lecture 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.

Lecture 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.

Lecture 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 demo: singlevaluedness

Lecture 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.

Lecture 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.

Lecture 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.

Lecture 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.

Lecture 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, ripplons, confinement.