Department Mathematik
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Mathematical Quantum Mechanics II (Summer 2026)

Phan Thành Nam, Dong Hao Ou Yang, Yaojun Wang

Homework, Moodle (ID: 46600, pass: MQM2026).

General Information

Goal: We study the mathematical methods for large quantum systems. We will discuss microscopic theory (many-body quantum mechanics), macroscopic theories (effective equations), and the connection between them.

In particular, we will focus on the Fermi gas, focusing on Pauli's exclusion principle, semiclassical analysis, Thomas-Fermi approximation and Hartree-Fock approximation. Previous background in functional analysis, especially operator theory on Hilbert spaces, will be helpful but not mandatory.

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

Time and place:
  • Lectures: Tuesday 8:15-10:00 (B004) and Friday 8:15-10:00 (B004).
  • Exercises: Friday 10:15-12:00 (B004).
  • Tutorials: Wednesday 10:15-12:00 (B132) and Friday 14:15-16:00 (B046).

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

Exams: There will be Midterm and Final exams.

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.

Contents of the lectures

14.4.2026. Introduction.

21.4. Chapter 1. Many-body quantum mechanics. Hilbert spaces. Operators. Tensor of Hilbert spaces. Symmetric (bosonic) and Anti-symmetric (fermionic) subspaces. Slater determinants.

21.4. Fermionic subspace (cont.). Anti-symmetrization. One-body density matrix. Pauli's exclusion principle.

24.4. Min-max principle. The ground state energy of the non-interacting fermionic system.

28.4. Chapter 2. Semiclassical approximation. Phase space approximation for the Schrödinger operator.

5.5. Lieb-Thirring inequalities: eigenvalue estimate and kinetic estimate. Cwikel–Lieb–Rozenblum (CLR) bound.

12.5. Coherent states. Weyl's law for the sum of eigenvalues of Schrödinger operator.

15.5. Dirichlet Laplacian on a bounded domain. Berezin-Li-Yau inequality. Weyl's law for the sum of eigenvalues. Tauberian argument. Weyl's law for individual eigenvalues.

19.5. Polya's conjecture. Weyl's law for trapping potential. Example of the harmonic oscillator.

22.5. Chapter 3. Thomas-Fermi theory. Density functional theory. Semiclassical and mean-field approximations. Thomas-Fermi functional.

29.5.