PHY604: Review of quantum mechanics

[Cr:4, Lc:2, Tt:2, Lb:0]

Classical vs. Quantum Mechanics, Simple 2-state QM system. Hilbert Spaces, Operators. Observables - Compatible Observables, Tensor Product Spaces, Uncertainty Relations. Position, Momentum and Translation. Eigenvalue Problems.
Time Evolution (Quantum Dynamics). Schroedinger, Heisenberg and Interaction Pictures; Energy-time Uncertainty, Interpretation of Wavefunction. Ehrenfest, Quantization, Path Integrals. Quantum Particles in Potential and EM Fields - Gauge Invariance, Aharanov-Bohm.
Angular Momentum: ${\mathrm SO}(3)$ vs. ${\mathrm SU}(2)$ . Lie Algebra and Representations of ${\mathrm SU}(2)$ . Spherical Harmonics. Addition of Angular Momenta. Tensor Operators and Wigner-Eckardt Perturbation Theory: Rayleigh-Schroedinger (Nondegenerate Time-independent) Perturbation Theory. Examples in Hydrogen Atom.
Symmetry groups in QM. Parity. Time reversal. Identical particles - permutations. Pauli exclusion. Central field approximation. Hartree equations. Scattering: Born approximation. Spherical waves. partial wave scattering. Low-energy scattering, bound states, resonances. Coulomb scattering.
Relativistic quantum mechanics: Dirac equation. Klein-Gordon equation. Relativistic particles and group theory. Solutions to Dirac: free particle, relativistic Hydrogen atom.