Advanced Graduate Quantum Mechanics

fizyka

Quantum mechanics, classical electrodynamics, statistical physics, or equivalent ones, and basic corses in mathematics and mathematical analysis.

mieszany: w sali i zdalnie
w sali
zdalnie

The corse discusses various selected topics in quantum mechanics typically not included in basic courses. The emphasis is put on physical aspects not on a mathematical rigor.

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.

Different book's chapters and review articles provided during the course.

A student should know about various extensions and applications of quantum mechanics in modern physical sciences.

A student should be able to solve basic problems illustrating discussed topics.

written colloquium and written and oral exams