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# Quantum Mechanics for Mathematicians

2020/2021
ENG
Instruction in English
8
ECTS credits

### Course Syllabus

#### Abstract

Within the framework of the discipline "Quantum mechanics for mathematicians", it is planned to present, in particular, the following topics in a form accessible to mathematics students: the algebra of observables in classical mechanics, states in classical statistical mechanics, physical bases of quantum mechanics, states in quantum mechanics, the Schrödinger equation, the Heisenberg commutation relations, coordinate and momentum representations, the interconnection between quantum and classical mechanics, the harmonic oscillator, the hydrogen atom, scattering of a one-dimensional particle by a potential barrier. Knowledge of classical mechanics is welcome.

#### Learning Objectives

• Formation and generalization of knowledge on quantum mechanics at the level of fundamental physical theory; mastering the mathematical apparatus of quantum mechanics; formation of the ability to apply theoretical knowledge in solving problems of quantum mechanics; the development of physical thinking; mastery of theoretical methods of cognition.

#### Expected Learning Outcomes

• To have a concept of the algebra of observables and states in classical mechanics. To know Liouville’s theorem, and two pictures of motion in classical mechanics. To understand the physical bases of quantum mechanics.
• To have a concept of the observables and states in quantum mechanics. To know Heisenberg uncertainty relations. Physical meaning of the eigenvalues and eigenvectors of observables. To understand two pictures of motion in quantum mechanics. To be able to write and solve Schrödinger equation. To be able to find stationary states.
• To know the Heisenberg commutation relations. To be able to pass from coordinate representations to momentum representations and back. To have a concept of eigenfunctions of the operators Q and P. To know the interconnection between quantum and classical mechanics. To be able to calculate the eigenfunctions and eigenvalues of the free one-dimensional particle and harmonic oscillator. To have a concept of the angular momentum of a three-dimensional particle.

#### Course Contents

• Elements of classical mechanics and physical bases of quantum mechanics
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
Observables. States in quantum mechanics. Heisenberg uncertainty relations. Physical meaning of the eigenvalues and eigenvectors of observables. Two pictures of motion in quantum mechanics. The Schrödinger 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. The general case of one-dimensional motion. The angular momentum of a three-dimensional particle. Scattering by a rectangular barrier.

#### Assessment Elements

• Домашние задания
• Итоговый устный опрос

#### Interim Assessment

• Interim assessment (1 module)
0.5 * Домашние задания + 0.5 * Итоговый устный опрос
• Interim assessment (2 module)
0.5 * Домашние задания + 0.5 * Итоговый устный опрос

#### Recommended Core Bibliography

• John Archibald Wheeler, & Wojciech Hubert Zurek. (1983). Quantum Theory and Measurement. Princeton University Press.