Quantum Mechanics II

Syllabus (under construction)

KKL - K. K. Likharev's Lectures on Quantum Mechanics
JJS - J.J. Sakurai's book
RRL - R. R. Liboff's book
MIT - MIT Open Courseware, Quantum Theory I
LL - L. D. Landau and E. M. Lifshitz, Quantum Mechanics
RS - R. Shankar, Principles of Quantum Mechanics
EM - E. Merzbacher, Quantum Mechanics
DJG - D. J. Griffiths, Introduction to Quantum Mechanics

# Date Read Topic
1. Tue, Feb 1 JJS 3.1-3.2 The theory of angular momentum. Translations. Rotations in three spatial dimensions. Generators of rotations. SU(2) algebra. Unitary transformations of a Hilbert space corresponding to rotations. Operators of angular momentum components. The simplest case of spin 1/2 system. Spintors, algebra of Pauli matrices.
2. Thu, Feb 3 JJS 3.6 Transformations of linear operators under rotations. Groups, Lie groups, Lie algebras, representation theory (some terminology). Infinitely dimensional case: orbital angular momentum.
3. Tue, Feb 8 JJS 3.5-3.6 Homework 11 is given. Eigenvalue problem for angular momenta. Ladder operators. (2j+1) dimensional representation. Matrix elements of angular momentum operators.
4. Thu, Feb 10 JJS 3.6-3.7 Matrix elements of angular momentum operators. Representations of rotation operator. Orbital angular momentum and spherical harmonics. Angular momentum in coordinate representation. Eigenfunctions of L^2 are spherical harmonics.
5. Tue, Feb 15 JJS 3.7 Homework 12 is given. Legendre polynomials and associated Legendre functions. Example of two spin-1/2 systems.
6. Thu, Feb 17 JJS 3.7 Example of spin and orbital angular momenta. Formal theory of angular momentum addition. Clebsh-Gordan coefficients and their properties. Addition of angular momenta. Example of two spin-1/2 systems.
7. Tue, Feb 22 JJS 3.4 Homework 13 is given. Density operator. Density matrix. Density operator. Time evolution of density operator. Density operator in statistical mechanics.
8. Thu, Feb 24 JJS 3.4, DJG 12.1, 12.2 Measurements in quantum mechanics. Entanglement, quantum computing, etc. Entanglement. Schmidt decomposition. Bell inequality.
9. Tue, Mar 1 JJS 3.10 Homework 14 is given. Bell inequality. Brief intro into Tensor operators. Cartesian and spherical tensors.
10. Thu, Mar 3 JJS 3.10, 4.1-3 Matrix elements of tensor operators. Wigner-Eckart theorem. Selection rules. Symmetries in Quantum Mechanics. Discrete symmetries. Parity.
11. Tue, Mar 8 JJS 4.3-4.4 Homework 15 is given. Polar and axial vectors, scalar and pseudoscalars. Even and odd wave functions. Parity of rotational states. Theorem of parity of eigenstates of H. Symmetric double-well potential. Parity selection rule. Other discrete symmetries (point groups, lattice translations).
12. Thu, Mar 10 JJS 4.4 Time-reversal symmetry. Time-reversal symmetry in classical physics. Antiunitary operators. Construction of time-reversal (anti)symmetry operator.
13. Tue, Mar 15 JJS 4.4 Homework 16 is given. Time-reversal symmetry and reality of wave functions. Time-reversal for spin 1/2. Kramers degeneracy.
14. Thu, Mar 17 JJS 5.1 Perturbation theory I. Time-independent case. Time-independent perturbation theory. Non-degenerate case. Example of two-level system. Perturbation theory expansion for non-degenerate case. Normalization of the wave function. Example: quadratic Stark effect.
15. Tue, Mar 22 JJS 5.2 Degenerate case. Secular equation. Example: linear Stark effect.
16. Thu, Mar 24 Midterm - one-hour exam (open book, you are allowed to use your notes, solutions to homework problems and a textbook of your choice). There will be three problems: 1) Angular momentum, 2) Discrete symmetries, 3) Time-independent perturbation theory.
17. Tue, Mar 29 JJS 5.3 Homework 17 is given. Fine structure of atomic terms Hydrogen atom. Estimate of the energy of the ground state. Bohr radius and Rydberg. Laguerre equation. Relativistic corrections to the Hamiltonian of a hydrogenlike atom.
18. Thu, Mar 31 JJS 5.3 Application of perturbation theory. Fine structure of atomic terms. Zeeman effect. Lande and Paschen-Back cases.
19. Tue, Apr 5 JJS 5.5, 5.6 Homework 18 is given. Paschen-Back case. Perturbation theory II. Time-dependent case. Interaction representation. Example of a two-level system. Dyson series. Transition probabilities.
20. Thu, Apr 7 JJS 5.6 Time-dependent perturbation theory. Constant perturbation. Sudden pertrubation.
21. Tue, Apr 12 JJS 5.6, 5.8 Homework 19 is given. Constant and harmonic perturbations. Fermi's Golden Rule. Energy shift and decay width.
22. Thu, Apr 14 EM Ch.7 Semiclassical (WKB) approximation. Quasiclassical approximation. Expansion in Planck's constant. Wave function and classical action. Criterion of quasiclassicity.
Tue, Apr 19 No classes, Spring Break.
Thu, Apr 21 No classes, Spring Break.
23. Tue, Apr 26 EM Ch.7 Connection formulas. Bound states. Bohr-Sommerfeld quantization condition.
24. Thu, Apr 28 LL 52-53
DJG Ch.10
Homework 20 is given. Double-well potential. Transmission through the barrier. Resonant transmission. Reflection above the barrier. The method of complex classical paths.
25. Tue, May 3 LL 52-53
JJS 2.5
The method of complex classical paths. Introduction to path integrals. Path integral representation of transition amplitudes. Path integral for phase space trajectories. Feynmann path integral.
DJG Ch.10
JJS Suppl.1
LL 53
SELF-STUDY. Adiabatic approximation. Adiabatic theorem. Dynamic and Berry's phases. Berry's (geometrical) phase. Landau-Zener transitions (LL 53).
26. Thu, May 5 JJS 6.1-6.2 Homework 21 is given. Identical particles and spin. The permutation symmetry. Symmetrization postulate. Bosons and fermions. Spin-statistics theorem. Two-electron system. Spin triplet and spin singlet: orbital and spin functions. Exchange density and exchange interaction.
27. Tue, May 10 JJS 7.1-7.2 Introduction to scattering theory. Scattering problem. Differential and total scattering cross sections in classical mechanics. Scattering amplitude. The Lippmann-Schwinger equation. Green's function.
28. Thu, May 12 JJS 7.2-7.3
The Born approximation. Example of Yukawa potential. Coulomb limit. Optical theorem. Scattering of slow particles.
JJS 7.5-7.6 SELF-STUDY. Method of partial waves and phase shifts.
Thu, May 19 FINAL EXAM: Room P-116, 11:15am-1:45pm The final is "closed books and notes".