MATH-495 / 10 credits

Teacher:

Language: English

Remark: pas donné en 2021-22


Summary

Quantum mechanics is one of the most successful physical theories. This course presents the mathematical formalism (functional analysis and spectral theory) that underlies quantum mechanics. It is simultaneously an introduction to mathematical physics and an advanced course in operator theory

Content

Keywords

Mathematical quantum theory; Fourier transform; spectral theory of unbounded operators; quantum observables; wave function; continuous and discrete spectrum; quantum dynamics

Learning Prerequisites

Required courses

Analyse I - IV, Algebre lineaire I et II, Analyse fonctionnelle I

Recommended courses

Analyse I - IV, Algebre lineaire I et II, Analyse fonctionnelle I, Physique quantique I et II

Important concepts to start the course

A firm background in mathematical analysis and rigorous proofs is required. This includes working knowledge of measure theory (especially L^p spaces) and basic functional analysis (especially Hilbert spaces). Familiarity with tools from partial differential equations, in particular Sobolev spaces and the Fourier transform, is not as important, but also useful.

Having previously completed a class on quantum mechanics is very helpful for understanding the motivations behind the results, but not strictly necessary.

Learning Outcomes

By the end of the course, the student must be able to:

  • Apply Fourier transform to describe free Schrodinger evolution
  • Formulate the basic notions of quantum mechanics rigorously
  • Distinguish spectral types and their dynamical implications

Transversal skills

  • Communicate effectively with professionals from other disciplines.
  • Access and evaluate appropriate sources of information.
  • Give feedback (critique) in an appropriate fashion.

Teaching methods

Four hours of lectures, two hours of exercises led by teaching assistant.

Expected student activities

Attend lectures and exercise sessions, read course materials, solve exercises.

Assessment methods

Graded homework sets and oral exam at the end of course.

Supervision

Office hours Yes
Assistants Yes
Forum No

Resources

Bibliography

In addition to the lecture notes mentioned below, the following textbooks are recommended for further reading.

  • E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Springer, 2001
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics: I Functional Analysis, second edition, Academic Press, 1980
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics: II Fourier Analysis, Self Adjointness, Academic Press, 1975

Ressources en bibliothèque

Notes/Handbook

The course will loosely follow an unpublished set of prior lecture notes that will be available on the course website.

In the programs

  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Mathematical quantum mechanics
  • Lecture: 4 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Mathematical quantum mechanics
  • Lecture: 4 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Mathematical quantum mechanics
  • Lecture: 4 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks

Reference week

 MoTuWeThFr
8-9     
9-10     
10-11     
11-12     
12-13     
13-14     
14-15     
15-16     
16-17     
17-18     
18-19     
19-20     
20-21     
21-22