Mathematical Quantum Mechanics II (Summer 2025)
Phan Thành Nam, Dong Hao Ou Yang
Homework - Moodle (Course ID: 39729, pass: MQM2025)
News: The midterm exam will take place on Friday June 27, 8:00-10:00.
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 Bose gas, focusing on the concept of Bose-Einstein condensation and Bogoliubov theory, as well as the Fermi gas, emphasizing Pauli's exclusion principle and semiclassical analysis. Previous background in functional analysis, especially operator theory on Hilbert spaces, will be helpful.
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: Tuesday 16:15-18:00 (B004).
References:
- Elliott H. Lieb, Jan Philip Solovej, Robert Seiringer, and Jakob Yngvason.
The Mathematics of the Bose Gas and its Condensation (Textbook 2005, Springer)
- Elliott H. Lieb and Robert Seiringer.
The Stability of Matter in Quantum Mechanics (Textbook 2010, Cambridge University Press)
- Jan Philip Solovej.
Many-body quantum mechanics (Lecture notes 2014)
- Phan Thanh Nam.
Mathematical Quantum Mechanics II (Lecture notes Summer 2020)
- Phan Thanh Nam.
Functional Analysis II (Lecture notes Winter 2020-2021)
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. You can bring your notes (lecture notes, homework sheets and tutorial materials) for these written exams. 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
25.4.2025. Introduction: Quantum mechanics, non-commutative operators, uncertainty principle, many-body systems, computational complexity, exclusion principle, bosons and fermions, Hartree/Hartree-Fock equations, Bogoliubov theory, superfluidity and superconductivity.
29.4. Chapter 1: Principles of quantum mechanics. Hilbert spaces and operators. States and Hamiltonians.
2.5. Ground states and Gibbs states. Tensor products of Hilbert spaces. Tensor products of operators.
6.5. Many-body quantum mechanics. Symmetrization: bosons and fermions. Spectrum of non-interacting systems.
9.5. Chapter 2: Schrödinger operators. Sobolev spaces. Uncertainty principle and Sobolev inequalities.
13.5. Lieb-Thirring inequality. Sobolev compact embedding theorem.
16.5. Applications to Schrödinger operators: boundedness from below, bound states and essential spectrum.
20.5. Chapter 3: Fock space formalism. Fock spaces. Creation and annihilation operators.
23.5. Canonical commutation relations (CCR) for bosons. Canonical anti-commutation relations (CAR) for fermions.
27.5. Second quantization.
30.5. Chapter 4: Reduced density matrices. One-body density matrices. Pauli exclusion principle for fermions. Ground state energy of the non-interacting Fermi gas.
3.6. Two-body density matrices. N-representability problem. Yang's inequality for fermions. Schur's Lemma.
6.6. Quantum de Finetti theorem for bosons.
10.6. Geometric localization method.
13.6. Extension of quantum de Finetti theorem for localized states.
17.6. Chapter 5: Hartree theory for Bose gases. The N-body mean-field Hamiltonian and the Hartree approximation. Convergence of the ground state energy in the mean-field regime.
20.6. Hellmann–Feynman argument. Bose-Einstein condensation of the ground state in the mean-field regime.
24.6. The Bose gas in the thermodynamic limit. Onsager's lemma and the energy in the high density limit. Remarks on the low density limit. Bose-Einstein condensation and off-diagonal long-range order.
27.6. Midterm exam
1.7. Chapter 6: Hartree-Fock theory for Fermi gases. The N-body mean-field Hamiltonian and the Hartree-Fock approximation.
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 Bose gas, focusing on the concept of Bose-Einstein condensation and Bogoliubov theory, as well as the Fermi gas, emphasizing Pauli's exclusion principle and semiclassical analysis. Previous background in functional analysis, especially operator theory on Hilbert spaces, will be helpful.
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: Tuesday 16:15-18:00 (B004).
References:
- Elliott H. Lieb, Jan Philip Solovej, Robert Seiringer, and Jakob Yngvason. The Mathematics of the Bose Gas and its Condensation (Textbook 2005, Springer)
- Elliott H. Lieb and Robert Seiringer. The Stability of Matter in Quantum Mechanics (Textbook 2010, Cambridge University Press)
- Jan Philip Solovej. Many-body quantum mechanics (Lecture notes 2014)
- Phan Thanh Nam. Mathematical Quantum Mechanics II (Lecture notes Summer 2020)
- Phan Thanh Nam. Functional Analysis II (Lecture notes Winter 2020-2021)
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. You can bring your notes (lecture notes, homework sheets and tutorial materials) for these written exams. 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.
Contents of the lectures
25.4.2025. Introduction: Quantum mechanics, non-commutative operators, uncertainty principle, many-body systems, computational complexity, exclusion principle, bosons and fermions, Hartree/Hartree-Fock equations, Bogoliubov theory, superfluidity and superconductivity.29.4. Chapter 1: Principles of quantum mechanics. Hilbert spaces and operators. States and Hamiltonians.
2.5. Ground states and Gibbs states. Tensor products of Hilbert spaces. Tensor products of operators.
6.5. Many-body quantum mechanics. Symmetrization: bosons and fermions. Spectrum of non-interacting systems.
9.5. Chapter 2: Schrödinger operators. Sobolev spaces. Uncertainty principle and Sobolev inequalities.
13.5. Lieb-Thirring inequality. Sobolev compact embedding theorem.
16.5. Applications to Schrödinger operators: boundedness from below, bound states and essential spectrum.
20.5. Chapter 3: Fock space formalism. Fock spaces. Creation and annihilation operators.
23.5. Canonical commutation relations (CCR) for bosons. Canonical anti-commutation relations (CAR) for fermions.
27.5. Second quantization.
30.5. Chapter 4: Reduced density matrices. One-body density matrices. Pauli exclusion principle for fermions. Ground state energy of the non-interacting Fermi gas.
3.6. Two-body density matrices. N-representability problem. Yang's inequality for fermions. Schur's Lemma.
6.6. Quantum de Finetti theorem for bosons.
10.6. Geometric localization method.
13.6. Extension of quantum de Finetti theorem for localized states.
17.6. Chapter 5: Hartree theory for Bose gases. The N-body mean-field Hamiltonian and the Hartree approximation. Convergence of the ground state energy in the mean-field regime.
20.6. Hellmann–Feynman argument. Bose-Einstein condensation of the ground state in the mean-field regime.
24.6. The Bose gas in the thermodynamic limit. Onsager's lemma and the energy in the high density limit. Remarks on the low density limit. Bose-Einstein condensation and off-diagonal long-range order.
27.6. Midterm exam
1.7. Chapter 6: Hartree-Fock theory for Fermi gases. The N-body mean-field Hamiltonian and the Hartree-Fock approximation.