## Advanced quantum mechanics for Ph. D. students

The course consists of two hour lecture and two hour problem class per week for two semesters.

The main idea of the course is to introduce the path integral approach to quantum mechanics as it has been done by R.P. Feynman in his famous book with A.R. Hibbs in 1965. Introductory part of the lecture is based on this book where we introduce basic ideas and perform practical calculations of quantum propagators for different systems. We also explore analogy with random walk and the Smoluchowski-Einstein approach to its theoretical description.

Special emphasis is put on the semiclassical approximation and the role of caustic properties of the classical motion in deriving the Bohr-Sommerfeld quantization formula. The interplay between fixed time and fixed energy actions is also discussed in this context.

Special emphasis is put on Euclidean transition amplitudes. We discuss instantons in quantum mechanics and calculate energy splitting in the dilute instanton gas approximation. Periodic potentials are also discussed.

Next, we discuss time evolution of the quantum system introducing time dependent perturbation theory in the language of path integrals. Bohm-Aharonov effect is discussed in this context.

A separate set of lectures concentrates on the scattering theory based on Lippmann-Schwinger equation. We introduce the notion of the cross-section, discuss optical theorem, eikonal approximation, partial waves, Breit-Wigner formula and finally inelastic scattering and form-factors.

We spend also some time on discussing some interpretational issues of quantum mechanics as the Schroedinger's cat, hidden variables and Bell's inequality, and quantum cryptography.

At the end we come back to the path integral in Euclidean space in the context of statistical physics. Following the Feynman book we discuss the polaron problem and, if time permits, Bose condensation and superfluidity.