Lectures on quantum mechanics for mathematics students

The algebra of observables in classical mechanics -- States -- Liouville's theorem, and two pictures of motion in classical mechanics -- Physical bases of quantum mechanics -- A finite-dimensional model of quantum mechanics -- States in quantum mechanics -- Heisenberg uncertainty relations -- Physical meaning of the eigenvalues and eigenvectors of observables -- Two pictures of motion in quantum mechanics. The Schrodinger equation. Stationary states -- Quantum mechanics of real systems. The Heisenberg commutation relations -- Coordinate and momentum representations -- "Eigenfunctions" of the operators Q and P -- The energy, the angular momentum, and other examples of observables -- The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics -- One-dimensional problems of quantum mechanics. A free one-dimensional particle -- The harmonic oscillator -- The problem of the oscillator in the coordinate representation -- Representation of the states of a one-dimensional particle in the sequence space [iota?]в‚‚ -- Representation of the states for a one-dimensional particle in the space D of entire analytic functions -- The general case of one-dimensional motion -- Three-dimensional problems in quantum mechanics. A three-dimensional free particle -- A three-dimensional particle in a potential field -- Angular momentum -- The rotation group -- Representations of the rotation group -- Spherically symmetric operators -- Representation of rotations by 2x2 unitary matrices -- Representation of the rotation group on a space of entire analytic functions of two complex variables -- Uniqueness of the representations D[subscript j] -- Representations of the rotation group on the space LВІ(SВІ). Spherical functions -- The radial Schrodinger equation -- The hydrogen atom. The alkali metal atoms -- Perturbation theory -- The variational principle -- Scattering theory. Physical formulation of the problem -- Scattering of a one-dimensional particle by a potential barrier -- Physical meaning of the solutions [psi]в‚Ѓ and [psi]в‚‚ -- Scattering by a potential center -- Motion of wave packets in a central force field -- The integral equation of scattering theory -- Derivation of a formula for the cross-section -- Abstract scattering theory -- Properties of commuting operators -- Representation of the state space with respect to a complete set of observables -- Spin -- Spin of a system of two electrons -- Systems of many particles. The identity principle -- Symmetry of the coordinate wave functions of a system of two electrons. The helium atom -- Multi-electron atoms. One-electron approximation -- The self-consistent field equations -- Mendeleev's periodic system of the elements -- Appendix: Lagrangian formulation of classical mechanics