# Quantum Mechanics

## Unit aims

The aim of the unit is to provide mathematics students with a thorough introduction to nonrelativistic quantum mechanics, with emphasis on the mathematical structure of the theory.

## Unit description

Quantum mechanics forms the foundation of 20th century and present-day physics, and most contemporary disciplines, including atomic and molecular physics, condensed matter physics, high-energy physics, quantum optics and quantum information theory, depend essentially upon it. Quantum mechanics is also the source and inspiration for various fields in mathematical physics and pure mathematics.

The aim of the unit is to provide mathematics students with a thorough introduction to nonrelativistic quantum mechanics, with emphasis on the mathematical structure of the theory. Additionally, in conjunction with other units, it should provide suitably able and inclined students with the necessary background for further study and research at the postgraduate level. Two relevant research fields, namely quantum chaos and quantum information theory are at present strongly represented in the Mathematics Department and in the Science Faculty as a whole.

## Relation to other units

This unit cannot be taken by students who have taken or are taking relevant Physics units at either level 2 or level 3. It is a prerequisite for Quantum Chaos and Advanced Quantum Theory (except for students taking the corresponding Physics units). It is a useful preparation for Quantum Information Theory as well as for the quantum mechanical parts of Statistical Mechanics.

## Learning objectives

At the end of the unit the student should:

- be familiar with the time-independent and time-dependent Schroedinger equations, and be able to solve them in simple examples
- be familiar with the notions of Hilbert space, self-adjoint operators, unitary operators, commutation relations, understand their relevance to the mathematical formulation of quantum mechanics and be able to use the notions to formulate and solve problems
- understand the probabilistic interpretation of quantum states, and basic aspects of the relation between classical and quantum mechanics
- understand the quantum mechanical description of angular momentum and spin

## Transferable skills

Expressing physical axioms mathematically and analysing their consequences.

## Syllabus

- motivation
- time-dependent and time-independent Schroedinger equations with characteristic examples including square wells and barriers (illustrating bound and scattering systems and tunneling) and the harmonic oscillator
- The mathematical structure of quantum mechanics, including Hilbert space as state space, observables as self-adjoint operators, Dirac notation, spectral representation of operators
- Time evolution
- Quantum mechanical measurement: postulates on the probabilistic interpretation of quantum states, expectation value and dispersion, measurement of position and momentum, the commutator, Heisenberg's uncertainty principle
- The harmonic oscillator in terms of raising and lowering operators
- Angular momentum: orbital angular momentum, raising and lower operators for angular momenta, spin

There may be minor changes to this syllabus.

## Reading and References

K. Hannabuss, *An Introduction to Quantum Theory*, Oxford 1997

Students may also find the following books interesting for further reading:

- C. J. Isham,
*Lectures on Quantum Theory*, Imperial College Press, 1995 - A. Peres,
*Quantum Theory: Concepts and Methods*, Kluwer, 1995

**Unit code:** MATH35500

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Unit director:** Dr Sebastian Muller

**Lecturer:** Dr Sebastian Muller

## Pre-requisites

Either Ordinary Differential Equations 2 or Applied Partial Differential Equations 2

## Co-requisites

None

## Methods of teaching

The unit consists of 30 lectures and 6 problem and revision classes. There will be lecture notes and weekly problem sheets.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.