Mathematical Quantum Mechanics 1

Module PH7006

This module is offered by Ludwig-Maximilians University Munich (LMU). It is available for TUM students only within a joint degree program (e. g. M. Sc. Quantum Science & Technology).

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information

PH7006 is a semester module in English language at Master’s level which is offered in winter semester.

This Module is included in the following catalogues within the study programs in physics.

Focus Area Theoretical Quantum Science & Technology in M.Sc. Quantum Science & Technology

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workload	Contact hours	Credits (ECTS)
270 h	90 h	9 CP

Responsible coordinator of the module PH7006 is Peter Müller.

Content, Learning Outcome and Preconditions

Content

This module introduces the basic elements of mathematical quantum mechanics. First, the mathematical basics of unbounded and self-adjoint operators (domain of definition, graphs, adjoints, spectrum, criteria for self-adjointness, spectral theorem, quadratic forms) will be discussed. Then Coulomb-Schrödinger operators, the essential spectrum, invariance under compact perturbations and the minimax principle will be presented. This is followed by elements of the theory of many-particle systems (density functional theory, second quantization) and its applications (e.g. Hartree-Fock approximation, superconductivity and superfluidity). At the end the basics of scattering theory (one- particle problems, the existence of wave operators) will be discussed.

Learning Outcome

Students become familiar with the foundations and essential methods of mathematical quantum mechanics. They are able to understand analytic methods and apply those to quantum mechanical questions. These notions are foundational to further advanced courses that will investigate these topics at greater depth.

Preconditions

No prerequisites in addition to the requirements for the Master’s program in Quantum Science and Technology. Familiarity with quantum mechanics is assumed, at the level of an introductory course from a Bachelor degree in physics. Basic notions of functional analysis can be helpful.

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

WS 2022/3 WS 2021/2 WS 2020/1

Type	SWS	Title	Lecturer(s)	Dates	Links
VO	4.0	Mathematische Quantenmechanik	Hainzl, C.	see LSF at LMU Munich	current
UE	2.0	Übungen zu Mathematische Quantenmechanik	Hainzl, C.	see LSF at LMU Munich	current

Learning and Teaching Methods

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week. The main teaching material is presented on the blackboard. Lectures are supplemented by weekly problem sets, deepening the understanding of core concepts through concrete calculations. Solutions to the problem sets are discussed in the exercise sessions. Participation in the exercise classes is strongly recommended, since the exercises are aids for acquiring a deeper understanding of the core tools of condensed matter many-body physics and field theory and for practicing to solve typical exam problems.

Media

Blackboard presentations, slides.

Literature

Standard textbooks on many-body theory, e.g.:

Elliott Lieb und Michael Loss “Analysis”
Gerald Teschl “Mathematical Methods in Quantum Mechanics”

Module Exam

Description of exams and course work

There will be a written exam of 180 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using conceptual questions and computational task.

For example an assignment in the exam might be:

Show that a Schrödinger Hamiltonian where the potential has some given decay properties has an infinite number of bound states.
Show that the atomic Thomas-Fermi-Energy converges to the true ground state energy in the limit of large nuclear charge.

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.

Exam Repetition

The exam may be repeated at the end of the semester.