Department Mathematik
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Mathematical Quantum Mechanics (Winter 2024-2025)

Phan Thành Nam (Math. Lecturer)
Robert Helling (Phys. Lecturer)
Long Meng (Exercises)
Charlotte Dietze (Tutorials)

Lecture notes, Homework, Moodle (ID: 37145, pass: MQM2425)

News

We plan to offer the retake exam on Friday, April 25, 2025. This will be an oral exam, starting from 2pm. If you are interested in taking it, please write to Phan Thanh Nam (nam@math.lmu.de) by the end of March 2025.

In case you want to receive your schein, please write to Long Meng to make an appointment.

You can review your final exam results on Tuesday, February 25, 2025, at 16:15 in room B045. If you missed this opportunity and still want to review your solutions, please write to Long Meng (meng@math.lmu.de) to arrange an appointment.

February 3-7: Lecture Tuesday, Tutorial Tuesday, Exercise Wednesday, Final Exam Friday 9:00-12:00 (B006).

The tutorial session on Friday holds online via zoom link (Meeting-ID: 676 0363 3948, code: 057765). Some material for the tutorial session: Notes Nov 22, Notes Nov 8, Notes Dec 6, Notes Dec 13, Notes Jan 10, Notes Jan 17, Notes Jan 24, Notes Jan 31.

The homework will be posted on every Wednesday.

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: Tuesday 8:30-10:00 (B005) and Friday 10:15-11:45 (B006).
  • Exercises: Wednesday 8:30-10:00 (B005).
  • Tutorials: Tuesday 16:15-18:00 (B045) and Friday 16:15-18:00 (A027).

References:
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: Midterm exam on December 17 (8:00-10:00) and Final exam on February 7 (9:00-12:00). 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.

Contents of the lectures

15.10.2024. Introduction. Chapter 1: Hilbert spaces and operators. Hilbert spaces. Inner product. Orthogonal projection. Riesz representation theorem. Dual space. Dirac bra-ket notation. Parseval theorem. L^2 and L^p spaces.

18.10. Unbounded operators. Spectrum. Adjoint operators. Symmetric and self-adjoint operators. An example of symmetric operators and self-adjoint extensions: the momentum operator on a bounded interval.

22.10. The lack of strong compactness in infinite dimensions. Weak convergence. Banach-Alaoglu theorem. Remarks on weak- and weak-* topologies. Principles of quantum mechanics. Real spectrum vs. self-adjointness.

25.10. Chapter 2: Spectral theorems. Compact operators. Spectral theorem for compact operators.

29.10. Schatten spaces. Trace class operators. Hilbert-Schmidt operators. Kernels of operators. Multiplication operators.

5.11. Continuous functional calculus for bounded self-adjoint operators. Riesz-Markov theorem. Spectral theorem for bounded self-adjoint operators.

12.11. Spectral theorem for bounded normal operators. Spectral theorem for unbounded self-adjoint operators. Remarks on functional calculus.

15.11. Chapter 3: Self-adjoint extensions. Closure of a symmetric operator. Closure of a quadratic form domain. Friedrichs' extension.

18.11. KLMN theorem. Kato-Rellich theorem. Remarks on family of self-adjoint extensions. Examples of self-adjoint extensions of Laplacian on (0,1).

22.11. Chapter 4: Schrödinger operators. The free Laplacian. Sobolev spaces. Weak derivatives via Fourier transform. Perron–Frobenius principle. Hydrogen atom.

26.11. Harmonic oscillator. Heisenberg uncertainty principle. Hardy inequality. Scaling argument.

29.11. Sobolev inequality. Sobolev compact embedding theorem.

3.12. Applications to Schrödinger operators: the self-adjointness by Kato-Rellich theorem, the boundedness from below and Friedrichs extension, and the compact resolvent for confining potentials.

6.12. Chapter 5: The algebraic approach. Motivation and definition of C*-algebras, examples. Gelfand’s theorem.

10.12. Functional calculus for normal observables, maps of C*-algebras, representations. The Weyl-algebra of canonical commutation relations, Stone-von Neumann theorem.

13.12. States on C*-algebras, examples. GNS-construction.

17.12. Midterm exam.

20.12. Proof of GNS construction. Xmas bonus Decoherence: Why does the quantum world appear classical at macroscopic scales?

7.1. 2025. Chapter 6: The discrete spectrum and essential spectrum. Weyl's criterion. Invariance of essential spectrum under relatively compact perturbations.

10.1. Min-max principle. Application to Schrödinger Operators.

14.1. Potential decay and the number of negative eigenvalues. Cwikel-Lieb-Rozenblum (CLR) inequality.

17.1. Chapter 7: Quantum dynamics. Schrödinger evolution equation. Strongly continuous one-parameter unitary group. Stone theorem.

21.1. Free Schrödinger dynamics. Long time behavior and dispersive estimates. RAGE theorem.

24.1. Further remarks on Stone theorem.

28.1. Wave operators. Cook method. Asymptotic completeness.

31.1. Concrete examples of the scattering problem.

4.2. General perspectives and final remarks.

7.2. Final exam.