What is the basis of quantum theory

Mathematical foundations of quantum mechanics summer semester 2015

Dates and rooms

  • Lecture: Tues. 4-6 p.m. in N24 / 131 & Thurs. 4-6 p.m. in N24 / 226

Lecture script

A script will be created to accompany the lecture. This is a preliminary version that will be expanded and revised parallel to the event with a time delay.

exam

The exam takes the form of an oral exam. For physicists, no further advance payment is required for admission to the examination apart from regular attendance at the event. Mathematicians can only have the course examined as 4 + 2 SWS if they also give a two-hour lecture on a supplementary topic.

content

In this strongly interdisciplinary course, the mathematical methods that are used to describe classical quantum mechanics are to be worked out precisely.

On the one hand, the event is aimed at mathematicians who want to learn the physical fundamentals of quantum mechanics and are interested in the applications of functional analytical methods in physics. On the other hand, the event is aimed at physicists who are interested in the mathematics behind quantum mechanics, which can only be touched on in the physical events.

We want to pay attention to a detailed application of the theory learned to concrete physical questions and quantum mechanical models as well as a clear representation of the mathematical methods. The lectures are therefore held alternately by the lecturers: in one half, physical applications and experiments are presented, in the other half the associated mathematical framework is worked out.

Specifically, we want to deal with the following topics:

  • Basic quantum mechanical phenomena
  • The postulates of classical quantum mechanics
  • Hilbert spaces & states in quantum mechanics
  • Sobolev spaces and Fourier transform
  • Bounded and unbounded operators on Hilbert spaces
  • Schrödinger operators
  • The spectral theorem for self adjoint operators & the quantum mechanical measurement process
  • The time evolution in quantum mechanics (Schrödinger equation) & unitary groups on Hilbert spaces
  • Distribution theory

requirements

The content of the basic lectures in Analysis and Linear Algebra is necessary, for example in the form of the courses Analysis I / II and Linear Algebra I for mathematicians or Advanced Mathematics I-III for physicists or other courses.

The content of the events is also helpful, if not necessary

  • Ordinary differential equations
  • Measure theory (for mathematicians)
  • Theory of Hilbert spaces, e.g. as part of the events Elements of Functional Analysis or Hilbert Spaces & Fourier Transformation (for mathematicians)
  • Quantum mechanics (for physicists)

Knowledge of these events is not expected and is provided in the lectures, but a first contact with some of the above topics would certainly be an advantage.

literature

Some of the literature can be viewed in the semester reserve.

Exam relevance

The lecture is assigned as follows in mathematics and physics.

  • Mathematics: Master, Pure Mathematics
  • Physics: Master, compulsory elective module

The course can also be heard as a minor course in the Master's Mathematics. In mathematics it can be examined as a 4 + 2 SWS course (for this, however, an additional lecture must be given as an examination requirement) and in physics as a two-hour course with a limited scope of material.

care, support

  • Lecturers:
    Prof. Dr. Wolfgang Schleich
    Dr. Stephan Fackler
    Dr. Kedar Ranade

Topics for the seminar

Suggested topics