Advanced Graduate Quantum Mechanics 1100-SZD-AGQM

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

6. Different formulations of quantum mechanics

- Schroedinger formulation
- Heisenberg formulation
- density matrix, pure and mixed states
- resolvent
- phase space formulation, Wigner quasi probability function
- path integrals
- Bohm theory
- Fock space and occupation number formalism

7. Measurement and interpretation problems in quantum mechanics

- Bohr
- von Neumann
- Everett
- Bohm

8. Entanglement states

- concept of entanglement, Bell states
- EPR paradox and its discussion
- no-cloning theorem
- teleportation algorithm
- Bell inequalities and experimental verifications
- progress and prospects in quantum computing

9. Quantum-classical correspondence

- signature of chaos in quantum systems
- random-matrix theory
- level statistics

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