Physics 302 2007 Lecture summaries and lecture notes.

These are the lecture notes for Physics 302 - Advanced Quantum Mechanics taught in Spring 2007.

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.

Readings are from Townsend: A modern approach to quantum mechanics


  1. Lecture 1: M 01/22/07, Why we need quantum mechanics. Introduction to Gaussian units. Reading: Appendix A pp 444-448
  2. Lecture 2: W 01/24/07, The Stern - Gerlach experiment. Four different experiments. Reading: Chapter 1 Section 1.1 pp 1-5, Section 1.2 pp 5-9
  3. Lecture 3: F 01/26/07 The quatum state vector and analysis of the four Stern Gerlach experiments. Chapter 1 Section 1.3 pp 9-13, Section 1.4 pp 13-15, Section 1.5 pp 15-18, Section 1.6 pp 18-21
  4. Lecture 4: M 01/29/07 Analyzing Stern Gerlach experiments using the quantum state vector. Freedom of representation and change of basis. Reading: Section 1.5 pp 15-18, Section 1.6 pp 18-21 Section 2.1 pp 24-28
  5. Lecture 5: W 01/31/07 Rotation operators. The identity and projection operators. Matrix representation of operators. Reading: Section 2.2 pp 28-35, Section 2.3 pp 36-41 Section 2.4 pp 41-45
  6. Lecture 6: F 02/02/07 Changing representation. The polarization of the photon. Reading: Section 2.5 pp 45-50, Section 2.6 pp 50-51, Section 2.7 pp 51-56
  7. Lecture 7 M 02/05/07 Action of rotations on vectors in lab and in Hilbert space. Matrix elements of components of angular momentum $\hat J_x$, $\hat J_y$, $\hat J_z$. An arbitrary unitary operation on a spin $\frac{1}{2}$ system can be made up of three rotations. Reading: Section 2.2 pp 28-35. Section 2.4 pp 41-45 Section 2.5 pp 45-50
  8. Lecture 8 W 02/07/07 Review of formalism. States $\leftrightarrow$ Hilbert space unit vectors, Observables $\leftrightarrow$ Hermitian operators etc. Expectation values. Projections and measurements. Reading: Section 2.8 pp 56-61 Section 2.3 pp 36-41
  9. Lecture 9 F 02/09/07 Completeness of basis and change of basis of operators. Schroedinger picture vs. Heisenberg picture. Non- commuting operators and angular momentum. Commuting operators. Reading: Section 2.4 pp 41-45
  10. Lecture 10 M 02/12/07 Commutation relations for generators of rotations. Defining angular momentum via commutation relations. Reading: Section 3.1 pp 64-69 Section 3.2 pp 69-70
  11. Lecture 11 W 02/14/07 Eigenvalues of the angular momentum operators. Reading: Section 3.3 pp 70-77
  12. Lecture 12 F 02/16/07 Raising and lowering operators. Matrix elements of raising and lowering operators. Reading: Section 3.3 pp 70-77 Section 3.4 pp 77-78
  13. Lecture 13 M 02/19/07 Commutation relations and eigenvalues of angular momentum operators. Reading: Section 3.2 pp 69-70, Section 3.3 pp 70-77
  14. Lecture 14 W 02/21/07 Commutation relations and Matrix Elements, Commutation relations and Uncertainty relations. Reading: Section 3.4 pp 77-78, Section 3.5 pp 78-80
  15. Lecture 15 F 02/23/07 Summary of angular momentum, examples for spin $1/2$ and spin $1$. Reading: Section 3.6 pp 80-85, Section 3.7 pp 85-87, Section 3.8 pp 88-90
  16. Lecture 16 M 02/26/2007 Commutation relations and the Uncertainty principle. Time dependence and the Hamiltonian. Reading: Section 3.5 pp 78-80 Section 4.1 pp 93-96 Section 4.2 pp 96-97
  17. Lecture 17 W 02/28/2007 A Spin-$\frac{1}{2}$ particle in a magnetic field. Reading: A Spin-$\frac{1}{2}$ particle in a magnetic field.
  18. Lecture 18 F 03/02/2007 Spin dynamics and magnetic resonance. Reading: Section 4.4 pp 104-108
  19. Lecture 19 M 03/05/2007 Basis states and addition of angular momentum for a system of two spin $\frac{1}{2}$ particles. Reading: Section 5.1 pp 120-122, {\em Section 5.2 pp 122-126 This section not covered in lecture}  Section 5.3 pp 126-131
  20. Lecture 20 W 03/07/2007 The EPR paradox and the Bell inequalities. Reading: Section 5.4 pp131-134, Section 5.5 pp 134-143.
  21. Lecture 21 F 03/09/2007 Class visit by Jerry Gollub - The EPR and Bohr discussion ``Can quantum mechanical description of reality be considered complete'' Einstein Podolsky Rosen Paradox
  22. Lecture 22 M 03/12/2007 Continuous variables: position eigenstates. The translation operator and linear momentum. Reading: 6.1    Position Eigenstates and the Wave Function    pp147-151, 6.2    The Translation Operator    pp151-153, 6.3    The Generator of Translations    pp153-156, 6.4    The Momentum Operator in the Position Basis    pp156-158 Relevant problems: 6.1, 6.2, 6.3, 6.7
  23. Lecture 23 W 03/14/2007 The relationship between the position and momentum bases. The fourier transform and the Heisenberg uncertainty principle. Reading:6.5    Momentum Space    pp158-160, 6.6    A Gaussian Wave Packet    pp160-164, 6.7    The Heisenberg Uncertainty Principle    pp164-166 Relevant problems: 6.4, 6.5, 6.6
  24. Lecture 24 F 03/16/2007 Simple properties of 1-D wave mechanics. Particle in a box. Reading: 6.8    General Properties of Solutions to the Schrodinger Equation in Position Space    pp166-171, 6.9    The Particle in a Box     pp171-177 Relevant problems: 6.8, 6.9 6.10, 6.11, 6.12, 6.13, 6.14, 6.15, 6.16
  25. Lecture 25 M 03/19/2007 Scattering in quantum mechanics. Incident, Reflected and Transmitted amplitudes from a step. Reading: 6.10    Scattering in One Dimension    177, 6.11    Summary    185, Relevant Problems 6.17, 6.18, 6.19, 6.20, 6.21
  26. Lecture 26 W 03/21/2007 Professor Eddie Farhi, MIT Center for Theoretical Physics class visit - presentation of ``A Quantum Algorithm for the Hamiltonian NAND Tree'' Reading: www.arxiv.org/quant-ph/0702144}, ``A Quantum Algorithm for the Hamiltonian NAND Tree'', E. Farhi, J. Goldstone, S. Gutmann  (Lecture given in recitation)
  27. Lecture 27 F 03/23/2007 The simple harmonic oscillator - operators and wavefunctions. Reading: 7.2    Operator Methods    196, 7.4    Matrix Elements of the Raising and Lowering Operators    201, 7.5    Position-Space Wave Functions    202, Relevant problems 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7
  28. Lecture 28 M 03/26/2007 Properties of the simple harmonic oscillator - zero point energy, classical correspondence, time dependence and symmetry. Reading: 7.6    The Zero-Point Energy    205, 7.7    The Classical Limit    207, 7.8    Time Dependence    208, 7.10    Inversion Symmetry and the Parity Operator    212, Relevant problems: 7.7, 7.8, 7.9, 7.11, 7.12
  29. Lecture 29 W 03/28/2007 Translations and momentum in three dimensions. Reading: 9.1    The Elements of Wave Mechanics in Three Dimensions    237 9.2    Translational Invariance and Conservation of Linear Momentum    241, 9.3    Relative and Center-of-Mass Coordinates 244. Relevant problems:  9.1, 9.2, 9.4, 9.5
  30. Lecture 30 F 03/30/2007 Centre of mass coordinates. Estimating Ground-State Energies Using the Uncertainty Principle. 9.4    Estimating Ground-State Energies Using the Uncertainty Principle 246, Relevant problems: 9.6, 9.7, 9.8
  31. Lecture 31 M 04/02/2007 Rotational invariance and conservation of angular momentum in three dimensions. Complete commuting observables for problems with spherical symmetry. Reading: 9.5 Rotational Invariance and Conservation of Angular Momentum 248 9.6 A Complete Set of Commuting Observables 250
  32. Lecture 32 W 04/04/2007 Angular momentum in position space (back to $\hat{\bf L}=\hat{\bf r}\times\hat{\bf p}$) and the spherical harmonics. Reading: 9.8    Position-Space Representations of L in Spherical Coordinates    260, 9.9    Orbital Angular Momentum Eigenfunctions    263 Relevant problems 9.12, 9.13, 9.14, 9.15, 9.16, 9.17, 9.18
  33. Lecture 33 F 04/06/2007 Solving the angular eigenvalue problem - the Spherical Harmonics Reading: 9.10    Summary    268 - Additional reading from Boas, Methods of Mathematical Physics.
  34. Lecture 34 M 04/09/2007 Reading: 10    Bound States of Central Potentials    274 10.1    The Behavior of the Radial Wave Function Near the Origin    274 10.2    The Coulomb Potential and the Hydrogen Atom     277, 10.3    The Finite Spherical Well and the Deuteron    288
  35. Lecture 35 W 04/11/2007 Reading:  10.4    The Infinite Spherical Well     292, 10.5    The Three-Dimensional Isotropic Harmonic Oscillator    296 10.6    Conclusion    302
  36. Lecture 36 F 04/13/2007 Reading: 11    Time-Independent Perturbations    306 11.1    Nondegenerate Perturbation Theory    306 11.2    An Example Involving the One-Dimensional Harmonic Oscillator    311
  37. Lecture 37 M 04/16/2007 Reading:  11.3    Degenerate Perturbation Theory    314 11.4    The Stark Effect in Hydrogen    316 11.6    Relativistic Perturbations to the Hydrogen Atom     322
  38. Lecture 38 W 04/18/2007 Reading: 11.7    The Energy Levels of Hydrogen, Including Fine Structure, the Lamb Shift, and Hyperfine Splitting    331
  39. Lecture 39 F 04/20/2007 Reading: 11.8    The Zeeman Effect in Hydrogen    334 11.9    Summary    335
  40. Lecture 40 M 04/23/2007 Reading: 14.6    Time-Dependent Perturbation Theory    417, 14.7    Fermi's Golden Rule    425
  41. Lecture 41 W 04/25/2007 Reading: 14.2    The Hamiltonian for the Electromagnetic Field    404, 14.3    Quantizing the Radiation Field    409, 14.4    The Properties of Photons     410, 14.5    The Hamiltonian of the Atom and the Electromagnetic Field    414
  42. Lecture 42 F 04/27/2007 Reading:14.8    Spontaneous Emission    430, 14.9    Higher-Order Processes and Feynman Diagrams    437