# MA4A7 Quantum Mechanics: Basic Principles and Probabilistic Methods

**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 MA3G7 Functional Analysis I, most notably the general framework of Hilbert spaces. We will use some notions from first-year Probability, which are taught in ST111 Probability A and ST112 Probability B (or equivalent modules). We will not assume any prior advanced knowledge of physics.

**Useful background: **It is also useful to have some familiarity with the Fourier transform. A module in which this is presented in a self-contained way is MA433 Fourier Analysis (in Term 1). Taking the module without having previously taken MA433 or an equivalent module is possible, although it would be a good idea to discuss this with the instructor beforehand. We will not directly use material from 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 SchrÃ¶dinger 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 SchrÃ¶dinger 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.