Mathematical Quantum Mechanics (Winter 2025-2026)

Phan Thành Nam (Math. Lecturer)
Robert Helling (Phys. Lecturer)
Alan Ramer dos Santos (Exercises)

Homework, Moodle (ID: 42326, Enrolment pass: MQM20252026)

News

The last exercise session takes place on Tuesday February 3. The final exam takes place on Friday February 6, starting from 10:00.

General Information

Goal: We study the fundamental mathematical concepts of quantum mechanics. In particular, we will discuss principles of quantum mechanics, self-adjoint operators, quadratic forms and Friedrichs extension, spectral theorems, Schrödinger operators, quantum dynamics, scattering theory, semiclassical analysis, and quantum entropy.

Audience : TMP-Master, Master students of Mathematics and Physics. Bachelor students will get "Schein" if pass the course.

Time and place:

• Lectures: Wednesday 8:30-10:00 (B005) and Friday 10:30-12:00 (B006).
• Exercises: Friday 16:15-18:00 (A027).
• Tutorials: Tuesday 14:15-16:00 (B006).

References:

• P.T. Nam and A. Scrinzi. Mathematical Quantum Mechanics (Lecture notes Winter 2018-2019)
• G. Teschl, Mathematical methods in quantum mechanics, AMS 2009.
• E. H. Lieb and M. Loss, Analysis, Amer. Math. Soc. 2001.
• Fredenhagen Bär, Quantum Field Theory on Curved Space-Times. Chapter 1
• Robinson Bratteli, Operator Algebras and Quantum Statistical Mechanics. Volume 1
• E. H. Lieb. Topics in Quantum Entropy and Entanglement (Lecture notes 2014).
• E. A. Carlen. Trace Inequalities and Quantum Entropy (Lecture notes 2009).

Exercises and Tutorials: There will be a homework sheet every week, which will be discussed in the exercise session. The tutorial sessions help to review the lectures and complementary materials.

Exams: There are written midterm and final exams. You can bring your notes (lecture notes, homework sheets and tutorial materials). Electronic devices are not allowed.

Grade: Final grade is determined by your total performance:

• You can get up to 100 points in the final exam.
• You can get up to 10 points in the midterm exam.
• You can get 1 point for every homework sheet.
  

You need 50 points to pass the course and 85 points to get the final grade 1.0.

Preparatory Course: October 6–10, 2025

Lectures: Monday-Friday, 10:15–12:00 (B047)

Exercises: Tuesday 16:15–18:00 (B047) and Friday 16:15–18:00 (B132)

Revision of mathematical background needed for Mathematical Quantum Mechanics. For the background in real analysis we follow Lecture notes (Chapter 1).

Contents of the lectures

15.10.2025. Introduction.

17.10. Chapter 1: Principles of quantum mechanics. Hilbert spaces. Operators on Hilbert spaces. Symmetric operators. Heisenberg's uncertainty principle. Stability question.

22.10. Hardy's inequality. Stability of the hydrogen atom.

24.10. Chapter 2: Operators on Hilbert spaces. Bounded and unbounded operators. Compactness. Weak convergence and Banach-Alaoglu theorem. Compact operators.

29.10. Spectral theorem for compact operators. Min-max principle for self-adjoint compact operators. Resolvent and spectrum. Multiplication operators.

31.10. Algebraic approach to quantum physics. Lecture notes

4.11. Basic properties of the spectrum of bounded self-adjoint operators.

7.11. Spectral theorem for bounded self-adjoint operators.

11.11. Algebraic approach to quantum physics (continued). References to C^* algebra literature: Fredenhagen Bär and Robinson Bratteli.

14.11. Spectral theorem for bounded normal operators. Spectral theorem for unbounded self-adjoint operators.

19.11. Applications of the spectral theorem. Chapter 3: Self-adjoint extensions. Closure and essential self-adjointness. Perturbation method and Kato-Rellich theorem.

21.11. Quadratic form and Friedrichs extension.

25.11. Algebraic approach to quantum physics (continued).

28.11. Algebraic approach to quantum physics (continued).

3.12. Chapter 4: Schrödinger operators. Sobolev space H^m(R^d). Weak derivatives via Fourier transform.

5.12. Sobolev inequalities. Stability of Schroedinger operators.

10.12. Lieb-Thirring inequality. Sobolev compact embedding.

12.12. Chapter 5: Weyl theory. Weyl's Criterion & Weyl sequences for the spectrum.

17.12. Discrete spectrum and essential spectrum. Weyl's Criterion for essential spectrum.

19.12. Min-max principle. Applications to Schroedinger operators.

7.1.2026. Chapter 6: Quantum dynamics. Time-dependent Schrödinger equation. Strong and weak solutions. Strongly continuous one-paramater unitary groups. Stone theorem.

9.1. Stinespring dilation theorem on characterization of completely positive maps.

13.1. Stinespring dilation theorem (continued). Free Schrödinger dynamics. Dispersive estimates.

16.1. RAGE theorem. Wave operators.

20.1. Cook's method. Duhamel formula. Asymptotic completeness.

23.1. Chapter 7: Quantum entropy. Von Neumann entropy. Relative entropy. Klein inequality. Partial traces and sub-additivity.

28.1. Gibbs variational principle. Peierls-Bogoliubov inequality. Golden-Thompson inequality.