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Mathematical Foundations of Quantum Mechanics

2020 – 2021, VIII, X semester

Course information

This course is a short introduction in the quantum mechanics designed specially for students of mathematical department. The course presumes from students to have no special prior knowledge of physics and simultaneously to have some advanced mathematical background including functional analysis, basic PDEs and theory of self-adjoint operators in Hilbert spaces. The particular attention in the course is devoted to the mathematical-friendly description of effects observed in physical experiments and their explanation in the mathematical language.

Another goal of this course is to provide some insight into relations of quantum mechanics with various branches of modern mathematics emphasizing the influence of quantum mechanics on the development of mathematics.

The rough plan of the course is following: main chapters of quantum physics and the place of quantum mechanics between them, experimental origins of quantum theory, observables and states, Heisenberg’s uncertainty relations, quantization, coordinate and momentum representations, Weyl relations and Stone-von Neumann theorem, quantum dynamics, Scr\” odinger equation, hydrogen atom,
spin, Pauli equation, system of identical particles, the symmetrization postulate and Pauli exclusion principle, Mendeleev periodic table from the point of view of quantum mechanics.

Prerequirements: in short mathematical introduction we recall briefly all necessary facts from the spectral theory of self-adjoint operator in Hilbert spaces. Nevertheless, students are strictly recommended to take prior this course some course/courses on the operator theory. For example, courses “Theory of Self-Adjoint Operators in Hilbert Space” and “Spectral Theory of Differential Operators, Parts I悠I” of
the BSc/MSc programs in Mathematics at SPbSU or equivalent are essential.


Course program


Lecturers

Associate Professor