Advanced Graduate Quantum Mechanics 1100-4AGQM

1. Symmetry in quantum mechanics
- symmetry operations
- T, P, C symmetries
- translational symmetry
- rotational symmetry
- conservation laws
- gauge symmetry
2. Symmetry breaking in quantum mechanics
- crystal lattice and symmetry breaking
- phonons as Goldstone modes
- superconductors, Anderson-Higgs modes and massive photons
3. Topology in quantum mechanics
- Aharonov-Bohm effect
- Landau levels and integer quantum Hall effect
- Berry phase
- topological insulators
- quantum number fractionalization
- SH polymer model
- zero modes and Majorana quasiparticles
4. Scattering and resonant states
- formal theory of scattering
- T and S matrix, symmetry, unitarity
- poles and branch cuts of S-matrix
- understanding of resonant states
5. Application of Rigged Hilbert spaces in quantum mechanics
- needs to extend the Hilbert spaces for unbounded, continuous operators
- spaces of physical, trial wave functions
- linear and anti linear distribution spaces
- rigorous interpretations of bra < | and ket | > Dirac states
- examples: free particles, resonant Gamow state

(*) In case of time shortage last topics will be canceled.