Mathematical Methods in Quantum Mechanics
A.Y. 2026/2027
Learning objectives
The course aims to provide mathematically advanced methods of functional analysis for the analysis of quantum systems.
Expected learning outcomes
By the end of the course, the student should understand and be able to apply rigorous functional analytic methods to the study of quantum systems.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Axiomatic framework of quantum mechanics
2. Banach spaces, Lebesgue spaces, and Hilbert spaces
3. Unbounded operators
4. Symmetric and self-adjoint operators; criteria for self-adjointness
5. Spectrum and resolvent
6. Kato-Rellich theorem and self-adjointness of Schrödinger operators
7. Strongly continuous unitary groups and Schrödinger time evolution
8. Symmetries and the quantum Noether theorem
9. Scattering theory
10. Spectral theorem
11. Exponential localization and regularity of bound states
12. Compact operators and stability of the essential spectrum
13. Spectral subspaces and the RAGE theorem
14. Perturbation theory and convergence
15. Many-Body Systems
16. Field Theoretic Methods
2. Banach spaces, Lebesgue spaces, and Hilbert spaces
3. Unbounded operators
4. Symmetric and self-adjoint operators; criteria for self-adjointness
5. Spectrum and resolvent
6. Kato-Rellich theorem and self-adjointness of Schrödinger operators
7. Strongly continuous unitary groups and Schrödinger time evolution
8. Symmetries and the quantum Noether theorem
9. Scattering theory
10. Spectral theorem
11. Exponential localization and regularity of bound states
12. Compact operators and stability of the essential spectrum
13. Spectral subspaces and the RAGE theorem
14. Perturbation theory and convergence
15. Many-Body Systems
16. Field Theoretic Methods
Prerequisites for admission
Basic knowledge of Real Analysis is required, in particular of Hilbert spaces and bounded operators; alternatively, knowledge of the contents of Mathematical Physics 3 or of a Quantum Mechanics course in the area of Theoretical Physics is required.
Teaching methods
The course is conducted primarily through in-person lectures.
Thematic in-depth studies are planned, based on assignments to be completed independently or in small groups, with active presentations by the students.
The course is classified as ALFA, with up to 10% online teaching. Online teaching may be used, in synchronous mode, primarily for lessons dedicated to introducing basic mathematical methods.
Thematic in-depth studies are planned, based on assignments to be completed independently or in small groups, with active presentations by the students.
The course is classified as ALFA, with up to 10% online teaching. Online teaching may be used, in synchronous mode, primarily for lessons dedicated to introducing basic mathematical methods.
Teaching Resources
- Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version
- Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version
- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
- Elliott H. Lieb, Michael Loss: Analysis.
- Gerald B. Folland: A Course in Abstract Harmonic Analysis
- Markus Haase: Lectures on Functional Calculus.
- Jan Philip Solovej: Many-body Quantum Physics, Lecture notes ESI 2014, https://web.math.ku.dk/~solovej/MANYBODY/
- Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version
- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
- Elliott H. Lieb, Michael Loss: Analysis.
- Gerald B. Folland: A Course in Abstract Harmonic Analysis
- Markus Haase: Lectures on Functional Calculus.
- Jan Philip Solovej: Many-body Quantum Physics, Lecture notes ESI 2014, https://web.math.ku.dk/~solovej/MANYBODY/
Assessment methods and Criteria
Assessment consists of an oral examination. During the examination, the student must demonstrate that they have understood the fundamental concepts, have acquired mastery of the mathematical methods, and are able to apply them to the solution of exercises.
An active presentation is also required, as specified in the teaching methods.
An active presentation is also required, as specified in the teaching methods.
MATH-04/A - Mathematical Physics - University credits: 6
Lessons: 42 hours
Professor:
Benedikter Niels Patriz
Shifts:
Turno
Professor:
Benedikter Niels PatrizProfessor(s)
Reception:
by appointment via e-mail
office 1024 (first floor, Via Cesare Saldini 50)