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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:

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)

Additional Resources