The condition for passing the course is to demonstrate sufficient knowledge of key subjects of theoretical physics at the master's degree level. Successful completion of the oral exam is a necessary condition for completing a master's degree.
A) Condensed matter theory:
Electrons in a periodic crystal potential. Bloch theorem. Born-van Kárman boundary conditions. Brillouin zone. Mean electron velocity in a crystal. Effective mass. Density of states. Approximation of nearly free electrons. Tight-binding method. Band structure. Electrons in a magnetic field. Landau levels. Lattice vibrations in harmonic approximation. Acoustic and optical modes in a linear chain with one and two atoms in a unit cell. Lattice vibrations of three dimensional lattice. Phonons. Specific heat of crystals. Optical properties of solids. Dielectric function. Optical conductivity. Superconductivity and effect on
physical properties of solids. Electron-phonon attractive interaction. Cooper pairs. Ground state and excited state of a superconductor. Itinerant and localized magnetism in solids. Magnons and spin waves in insulators.
B) Phase transitions and critical phenomena:
Phase equilibrium and phase transitions. Classical (Ehrenfest) and modern classification of phase transitions. Landau description of phase transitions: order parameter and symmetry breaking at continuous phase transitions. Critical indices and universality. Basic microscopic models of magnetic phase transitions: Heisenberg and Ising model. Exact solution of a one-dimensional Ising model in an external magnetic field. Mean (molecular) field approximation for Ising model. Phenomenological Landau theory of phase transitions. Tricritical point.
C) Quantum field theory:
Classical and quantum fields - general definition. Lagrange formalism for classical fields. Euler equations for fields. Symmetry and conservation laws. General dynamic invariants. Energy-momentum tensor. Free classical scalar (real and complex) field, Klein-Gordon equation. Dynamic invariants for scalar fields. Free classical electromagnetic field, Maxwell equations in covariant form. Dynamic invariants for electromagnetic fields. Free spinor field, Dirac equation. Dynamic invariants for a spinor field. Quantization of classical free fields, heuristic approach and general rules. Scalar field quantization. Spinor field quantization. Electromagnetic field quantization as an example of quantization of fields with constrains. Interacting fields, basic rules for introducing coupling members into Lagrangians. Local calibration invariance, minimal interaction of spinor and electromagnetic fields. Lagrangian of quantum electrodynamics. The concept of N -, S - and T - products of quantum-field operators. Wick theorem for the N-product. Wick theorem for T-product. Evolution operator, S-operator and S-matrix. Green functions as vacuum means of T products of free and interacting quantum fields. Generating functional of Green functions. Feynman diagram technique, general rules for graph construction. General rules for calculation of Green functions using perturbation theory. Compton scattering: calculation of the S matrix and the effective cross section for non-polarized particles in the leading order approximation.