Lecturer: Vedran Sohinger

Term(s): Term 2

Status for Mathematics students:

Commitment: 30 lectures

Assessment: 3 hour examination (100%)

Formal registration prerequisites: None

Assumed knowledge: In this module, we will assume the content of MA3G8 Functional Analysis II, although ideas from this module could be useful.

Synergies: The following modules could be helpful to take concurrently:

• MA4J0 Advanced Real Analysis
• MA4L2 Statistical Mechanics (We will not directly use any results from either module)

Content:

Quantum mechanics is one of the most successful and most fundamental scientific theories. It provides mathematical tools capable of describing properties of microscopic structures of our World. It is fundamental to the understanding of a variety of physical phenomena, ranging from atomic spectra and chemical reactions to superfluidity and Bose-Einstein condensation.

In the lectures we will discuss mathematical foundations of quantum theory: This includes the concepts of mixed and pure states, observables and evolution operator, a wave function in Hilbert space, the stationary and time-dependent Schrödinger equations, the uncertainty principle and the connections with classical mechanics (Ehrenfest theorem).

We will give simple, exactly soluble examples of both time-dependent and time-independent Schrodinger equations. We will also touch some more advanced topics of the theory.

Aims:
To introduce the basic concepts and mathematical tools used in quantum mechanics, preparing students for areas which are at the forefront of current research.

Objectives:
The students should obtain a good understanding of the basic principles of quantum mechanics, and to learn the methods used in the analysis of quantum mechanical systems.

Books:

Stephen J. Gustafson, Israel M. Sigal: Mathematical Concepts of Quantum Mechanics, Springer Universitext 3rd Edition 2020.

Gerald Teschl: Mathematical Methods in Quantum Mechanics: with Applications to Schrödinger Operators, American Mathematical Society, Graduate Studies in Mathematics 157, Second Edition 2014.

Michael Reed, Barry Simon: Methods of Modern Mathematical Physics (Volume 1), Academic Press.

Elliott H. Lieb, Michael Loss: Analysis, American Mathematical Society, Graduate Studies in Mathematics 14, Second Edition 2001.

Additional Resources