# MA4A7 Quantum Mechanics: Basic Principles and Probabilistic Methods

**Lecturer:** Vassili Gelfreich

** Term(s):** Term 2

**Status for Mathematics students:**

**Commitment:** 30 lectures

**Assessment:** 3 hour examination (100%)

**Prerequisites:** There are no strict prerequisites. But knowledge of Partial differential equations and, in some parts, Functional Analysis, will be helpful.

**Leads To: **

**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:**

**S.J. Gustafson, I.M. Sigal, Mathematical concepts of quantum mechanics, Springer, 2011.**

**A. Messiah, Quantum mechanics, Dover, 1999.**

**L. D. Faddeev, O. A. Yakubovskiĭ, Lectures on quantum mechanics for mathematics students. Student Mathematical Library, 47. American Mathematical Society, Providence, RI, 2009, 234 pp**

**W.G. Faris, Outline of Quantum Mechanics, in Entropy and the Quantum, Contemp. Math. 529, 1-52 (2010)**

**J. Fröhlich, B. Schubnel, Do we understand quantum mechanics - finally? (2012)**