Mathematical Foundations of Quantum Mechanics Summer Term 2015

• Lecture: Tuesday 16:00-18:00 in N24/131 & Thursday 16:00-18:00 in N24/226

The following preliminary lecture notes accompany the lectures and will be created along the lectures with some weeks delay.

• Preliminary incomplete proofread version of the lecture notes for the mathematical part (version of May, 6)
• Complete not proofread version of the lecture notes for the mathematical part
• Preliminary version of the lecture notes for the physics part (version of June, 21)

There will be an oral exam at the end of the semester. For master students in physics there are no requirements for the exam except for the regular attendance of the lectures. Master students in mathematics can only take the oral exam for 9 credit points if they give a talk on a topic closely related to the content of the lecture. According to the examination regularations, if desired, physicists can also get credit points (without grades) for the attendance of the lectures without an oral exam.

In this interdisciplinary course we want to present in a mathematical rigorous way the mathematical methods used in the modern formulation of quantum mechanics.

On the hand hand the course is aimed at mathematicians who are interested in the applications of functional analytical methods in physics and want to learn the fundamentals of quantum mechanics. On the other hand the course is aimed at physicists who are interested in the mathematics used in the treatment of quantum mechanics.

The focus of the lecture is twofold: we want to present in detail quantum mechanical models and the outcome of foundational experiments in one half of the lecture, whereas the other half covers a rigorous mathematical treatment of the following topics:

• The postulates of classical quantum mechanics
• Hilbert spaces and states in quantum mechanics
• Sobolev spaces and the Fourier transform
• Bounded and unbounded linear operators on Hilbert spaces
• Schrödinger operators
• The spectral theorem for self-adjoint operators and measurement in quantum mechanics
• Time evolution in quantum mechanics (Schrödinger equation) and unitary groups on Hilbert spaces
• Theory of distributions

For the lecture basic knowledge in real variable calculus and linear algebra is assumed. Moreover, mathematicians surely profit from a previous exposure to the elements of Lebesgue integration and basic Hilbert space theory, whereas physicists would profit from a previous introductory course in quantum mechanics.

Some of the literature given below can be found in the semester apparatus in the university library.

• Robert Denk: Lecture notes Mathematische Grundlagen der Quantenmechanik
• Brian C. Hall: Quantum Theory for Mathematicians
• Leon. A. Takhtajan: Quantum Mechanics for Mathematicians
• Gerald Teschl: Mathematical Methods in Quantum Mechanics - With Applications to Schrödinger Operators

• Mathematics: Master, Pure Mathematics
• Physics: Master

Mathematicians can take an oral exam covering the full lecture for 9 credit points (with the additional prerequisite that they give an additional talk on a topic closely related to the contents of the lectures), physicists can take on oral exam covering half of the lecture (for fewer credit points).