﻿ Skoltech Center for Advanced Studies | theoretical minimum on Mathematical Physics

# theoretical minimum on Mathematical Physics

### Mathematics

• Derivatives and integrals (ability to differentiate and integrate).
• Linear algebra: vector spaces, direct sums, tensor products, factor-spaces. Systems of linear equations and matrices. Diagonalization of matrices, finding eigenvalues.
• Vector analysis: vector and tensor fields, gradient, divergency, rotor.
• Multidimensional integrals. Stokes formula.
• Notion of generalized functions (distributions). Delta function and its main properties.
• Ordinary differential equations (standard methods of solution).
• Partial differential equations. Change of variables. Methods of solution: method of characteristics, separation of variables, Fourier transform (for linear equations).
• Equations of mathematical physics (Laplace equation, heat equation, wave equation, Klein-Gordon equation,…). Method of Green functions. Waves in 1, 2 and 3 dimensions. Boundary value problems for the Laplace operator.
• Theory of functions of complex variable: holomorphic and meromorphic functions, Cauchy formula, integrals of Cauchy type, Sokhotsky formulas, evaluation of integrals by means of residues, main properties and examples of conformal maps, Riemann surfaces of functions.
• Asymptotical methods of evaluation of integrals: stationary phase, Laplace method, steepest descent method.
• Hypergeometric equation, its singularities and behavior of solutions near singularities. Degenerations of hypergeometric functions.
• Elliptic integrals. Definition and main properties of elliptic functions.
• Basics on group theory: normal subgroups,homomorphisms, symmetric groups, representations of finite groups.
• Notion of Lie groups and algebras, rotation group and SU(2), their finite dimensional representations.
• Basic notions of differential geometry: curvilinear coordinates, metric, geodesics, Christoffel symbols, curvature tensor.
• Differential forms, their integration, de Rham differential, definition of de Rham cohomology.
• Hyperbolic (Lobachevsky) geometry and its main models. De-Sitter and anti-De-Sitter spaces.
• Definition of homotopy and homology groups, simplest examples.
• Functional analysis: Lebesgue integral, Banach spaces, Hilbert space, unitary and selfadjoint operators, spectral theorem for bounded selfadjoint operators.
• Probability theory: random variables, their distributions, moments, characteristic functions, Chebyshev inequality, law of large numbers, central limit theorem.

### Mechanics and hydrodynamics

• Lagrangian mechanics: Lagrange function, equations of motion. Principle of least action. Symmetries and conservation laws. Integration of one-dimensional motion in a potential. Motion in centrally-symmetric field and Kepler problem.
• Hamiltonian mechanics: Legendre transformation, Hamilton function. Canonical Hamilton equations, Poisson brackets. Integrals of motion. Canonical transformations. Hamilton-Jacobi equation.
• Rotation of rigid body: Euler equations, integrals of motion.
• Basic principles and notions of hydrodynamics: continuity equation, equation of motion for ideal fluid (Euler equation), viscous fluids, Navier-Stokes equation.
• Basic notions of Special Relativity. Relativity principle. Minkovsky space. 4-vectors, Lorentz transformations. Relativistic kinematics. Action for relativistic particle. Conservation laws for energy and momentum in Special theory of Relativity.

### Classical field theory and gravity

• Classical field theory. Noether theorem and conserved currents. Energy-momentum tensor for scalar field. Free and interacting fields. Free field as a system of oscillators.
• Classical relativistic electrodynamics: gauge-invariant action and Maxwell equations. Energy-momentum tensor for electromagnetic field. Electromagnetic waves.
• Examples of classical field theories: Landau-Ginzburg, Liouville and и sine-Gordon (in two dimensions).
• Simplest soliton solutions of nonlinear partial differential equations.
• Gravitational field in relativistic mechanics. Equivalence principle. Coordinate systems in general theory of relativity. Relativistic particle in gravitational field. Limiting transition to Newton theory of gravity.
• Action for gravitational field. Einstein equations.
• Centrally-symmetric gravitational field. Schwarzschild solution. Black holes, event horizon.

### Quantum mechanics

• Basic notions of quantum mechanics: space of states, superposition principle, measurement, probability interpretation, wave function, Hamilton operator, energy spectrum, coordinate and momentum operators, uncertainty relation. Interpretation of quantum mechanics by means of path integral (according to Feynman).
• Schrodinger equation and its main properties. Solution of Schrodinger equation in one dimension. Discrete and continuous spectrum. Finding of energy levels of a particle in a potential well (simple examples). Scattering, finding of scattering coefficient (simple examples). Reflectionless potentials.
• Harmonic oscillator. Finding of spectrum and eigenfunctions of the oscillator by different methods. Creating and annihilation operators. Coherent states.
• Motion in centrally-symmetric field. Motion in Coulomb field. Separation of variables in the two-body problem. Hydrogen atom.
• Limiting transition to classical mechanics. Quasiclassical approximation.
• Penetration through potential barrier (tunnel effect). Bohr-Sommerfeld quantization rule.
• Basic methods of perturbation theory. Secular equation.
• Schrodinger equation in magnetic field. Energy levels of a charged particle in homogeneous magnetic field (Landau levels).
• Angular momentum. Adding angular momenta and connection with representation theory of rotation group.
• Notion of spin. Connection with representation theory of SU(2). Pauli matrices.
• Principle of identity of identical particles. Notion of second quantization. Bose and Fermi statistics.

### Statistical physics

• Basic notions of probability theory and random processes: probability distribution, independence and correlations, mean value, variance. Gaussian distribution. Poisson distribution.
• Microcanonical, canonical and grand canonical ensembles. Partition function.
• Notions of entropy and temperature in statistical physics and thermodynamics. Free energy and thermodynamic potentials.
• Ideal gas. Equation of state of ideal gas. Maxwell distribution.
• Fermi and Bose gases of elementary particles. Fermi and Bose distributions.
• Problem of radiation of absolutely black body. Planck’s formula.
• One-dimensional Ising model. Method of transfer matrix.
• Phase transitions and their classification. Critical point. Scale invariance.

### Literature

1. L.Landau and E.Lifshitz, Theoretical Physics, volumes I, II, III, V.
2. B.Dubrovin, S.Novikov, A.Fomenko, Modern Geometry.
3. M.Lavrentiev, B.Shabat, Methods of theory of functions of complex variable.
4. M. Reed and B. Simon, Methods of modern mathematical physics vol. I.
5. L.B. Koralov and Ya. G. Sinai, Theory of probability and random processes.
6. E.Whittaker and G.Watson, A course of modern analysis.