Mathematical Quantum Mechanics II (Summer 2020)

Phan Thành Nam, Dinh-Thi Nguyen

Final exam (9:00-13:00, 29.7.2020)

Online Lectures:

The lectures will be streamed live with ZOOM.

Join Zoom Meeting (ID: 943-8498-7521, PW: 787534).

Lecture notes - Homework Sheets - Uni2work

General Information

Goal: We study the mathematical foundation of many-body quantum mechanics.

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

Time:

• Lectures: Tuesday and Friday, 8:30-10:00.
• Exercises: Friday, 10:00-12:00.

Exercises: There will be a homework sheet every week. The solutions to the homework will be discussed in the exercise section. Doing the homework is the best way to learn the course's materials and to prepare for the final exam.

Grade: Final grade is determined by your total performance:

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

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

Exams: You can use your notes (lecture notes, homework sheets and tutorial materials).

References:

• J.P. Solovej, Many-body quantum mechanics, Lecture notes 2014.
• E. H. Lieb and R. Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, 2009.
• M. Reed and B. Simon, Methods of modern mathematical physics, Volume I-IV.

Preliminary contents of the course

• Principles of quantum mechanics.
• HVZ theorem and basic spectral properties.
• Mean-field approximation and Hartree theory.
• Fock space formalism.
• Quasi-free approximation and Bogoliubov theory.
• Semiclassical approximation, Hartree-Fock and Thomas-Fermi theory.

Contents of the lectures

21.4.2020. Chapter 1: Principles of quantum mechanics. Hilbert spaces and operators. Principles of quantum mechanics. Tensor product. Many-body Hamiltonian. Identical particles. Particle statistics: bosons and fermions.

24.4. Chapter 2: Schrödinger operators. Review of discrete spectrum, essential spectrum, Weyl's theory, Min-max principle, Sobolev's inequalities, IMS formula. Schrödinger operators with trapping potentials.

28.4. Schrödinger operators with vanishing potentials. Kato theorem (self-adjointness). HVZ theorem (essential spectrum).

5.5. Atomic Hamiltonian. Ionization problem: how many electrons that a nucleus can bind?. Zhilin theorem (existence). Lieb theorem (non-existence).

8.5. Chapter 3: Hartree theory. Introduction to Hartree theory. The existence of Hartree minimizers: trapping potentials and vanishing potentials. Concentration-compactness method. Binding inequality and strict binding inequality.

12.5. The existence of Hartree minimizers (cont.): translation-invariant case. Concentration-compactness lemma: weak convergence up to translations. Application to Choquard-Pekar Problem.

15.5. Hartree equation. Regularity of Hartree minimizer.

19.5. Diamagnetic inequality. Yukawa potential. Harnack's inequality. Strict positivity of Hartree minimizer. Unique ground state of the linearized equation. Positive-type potentials. Convexity and the uniqueness of Hartree minimizer.

22.5. Chapter 4: Validity of Hartree approximation. Reduced density matrices. Hoffmann-Ostenhof inequality. Onsager's lemma.

26.5. Proof of convergence to Hartree energy.

29.5. Bose-Einstein condensation. Proof of convergence to Hartree minimizer.

5.6. Short-range interactions. Convergence to the Gross-Pitaevskii energy and minimizer.

9.6. Chapter 5: Fock space formalism. Bosonic Fock space. Creation and annihilation operators. Canonical commutation relations. Method of second quantization.

12.6. Generalized one-body density matrices. Weyl operators. Coherent states.

16.6. Gaussian states. Wick's theorem. Quasi-free states.

19.6. Chapter 6: Bogoliubov theory. Bogoliubov heuristic argument. Example for the homogeneous gas.

23.6. Bogoliubov transformations. Examples of one and two dimensions. Necessary and sufficient conditions for the existence of Bogoliubov transformation.

26.6. Diagonalization of block operators: fermionic case.

30.6. Diagonalization of block operators: bosonic case.

3.7. Characterization of quasi-free states.

7.7. Diagonalization of quadratic Hamiltonians.

10.7. Chapter 7: Validity of Bogoliubov approximation. Bogoliubov Hamiltonian. A unitary operator implementing c-number substitution.

14.7. Transformed operator. Operator bounds on truncated Fock space.

17.7. Improved condensation. Derivation of Bogoliubov excitation spectrum.

21.7. Extension to singular potentials. Bogoliubov Hamiltonian with singular potentials.

24.7. Modified operator bound on truncated Fock space. IMS localization on particle number operator. Validity of Bogoliubov excitation spectrum with singular potentials.

28.7. Short range potentials. Scattering length. Derivation of the Gross-Pitaevskii equation. Slides.