The purpose of the course is to introduce students to the basic concepts and theoretical methods used in problems of modern quantum mechanics and solid state physics.
The course is divided into two parts. In the first part, we will study in detail theoretical methods for solving single-particle problems in quantum mechanics: the strong-binding approximation for finding the energy spectrum of crystals, various methods of time-dependent perturbation theory for describing the optical properties of quantum systems. In addition, we will be introduced to the concept of topology in quantum mechanics.
The second part is devoted to the basic method of studying many-particle interactions in quantum mechanics. Using examples of the simplest problems, we will become familiar with extremely effective theoretical methods, such as the mean field method, the diagram technique, and the basics of the method of functional integrals.
Part I. Single-particle problems in quantum mechanics
Approaching strong connection. Spectra of crystals
Time-periodic disturbances (Floquet method)
The concept of topology in quantum mechanics. Hall effects in quantum mechanics (ordinary, anomalous, spin). Topological invariants.
Density matrix. Pure and mixed quantum states.
Temporary representations in CM. Representation of Schrödinger, Heisenberg and interaction
Linear response theory. Kubo formula. Electrical and magnetic susceptibility of a quantum system
Part II. Introduction to many-particle QM methods.
Secondary quantization. Formalism of filling numbers. Wave functions of a many-particle system and operators in the representation of occupation numbers.
Mean field method. Phase transitions.
Basics of diagramming technology. Partial sum method. Hartree-Fock and random phase approximation for interacting electrons.
Method of functional integrals. Coherent states. Harmonic oscillator in the language of functional integrals. Stratanovich transformation. Anharmonic oscillator.