# Quantum Chaos

## Unit aims

At the end of the unit you will comprehend the central ideas behing Quantum Chaos and have an understanding of the most important issues of some topics of current research in the field.

## Unit description

Many systems in nature are chaotic, i.e., their classical time evolution depends sensitively on the initial conditions. This has important consequences for their quantum behaviour. At microscopic length scales, the chaotic dynamics of the corresponding classical system manifests itself in the behaviour of the eigenfunctions and of the energy levels of the quantum Hamiltonian. For example, when the classical motion is regular the eigenvalues of the quantum system appear as a sequence of uniformly distributed random numbers, while if the dynamics is chaotic they manifest a more rigid structure and tend to repel each other.

The course will discuss the main features of the spectra and eigenfunctions of quantum Hamiltonians whose classical limit is chaotic. We will introduce the most important mathematical techniques used to study these systems, such as the Gutzwiller trace formula. The unit will also include the main ideas behind two of the most important areas of research in the subject: the random matrix theory conjecture and the problem of quantum ergodicity. Important examples systems will be billiards. On a billiard table, a ball moves on a straight line and is reď¬‚ected at the boundary. If the shape of the billiard table is irregular the classical motion of the billiard ball becomes chaotic and the quantum mechanical energy levels of the system display level repulsion.

## Relation to other units

The unit requires basic knowledge in quantum mechanics. The Quantum Mechanics unit in the Mathematics department or its equivalent in Physics are prerequisites.

Some ideas discussed are related to topics presented in the level 3 unit Random Matrix Theory. Units dealing with classical chaos are "Applied Dynamical Systems" as well as "Dynamical Systems and Ergodic Theory" (from the viewpoint of Pure Mathematics) and "Nonlinear Dynamics and Chaos" (in Engineering Mathematics). Moreover there are connections to Mechanics 2/23 and Mathematical Methods. The semiclassical approximations in this unit are examples for asymptotic approximations studied in the Asymptotics unit and have alternative derivations using the path integral method introduced in Advanced Quantum Theory. All relevant material connected to these units will be introduced in a self-contained way.

## Learning objectives

At the end of the unit the student should:

• Be familiar with important classical properties of chaotic systems (hyperbolicity, ergodicity) as well as their consequences in quantum mechanics (quantum ergodicity, universal spectral statistics).
• Understand and be able to apply the techniques used to connect quantum mechanics and classical mechanics (stationary-phase approximations, Gutzwiller's trace formula).
• Understand how the statistics of energy levels can be characterised, how it is connected to random matrix ensembles, and how Gutzwiller's trace formula and the diagonal approximation can be used to explain universal spectral statistics.
• Be able to apply the underlying ideas to solve typical problems in quantum chaos.

## Transferable skills

• Clear, logical thinking.
• Problem solving techniques.
• Assimilation and use of complex and novel ideas.

## Syllabus

0. Introduction [~1 lecture]

1. Classical chaos [~3 lectures]
• Hyperbolicity: sensitive dependence of the classical motion on initial conditions, stability matrices, Lyapunov exponent.
• Ergodicity: trajectories fill the available space "almost uniformly".
• Example: chaotic billiards.
2. Quantum ergodicity [~1 lecture]
• Features of quantum mechanical wavefunctions in systems that are classically chaotic.
3. The propagator [~4 lectures]
• The propagator describes the time evolution of a quantum system.
• Definition and properties.
• Examples: free particle, system with a single wall.
• Van Vleck approximation for the propagator of chaotic billiards.
4. Stationary-phase approximation [~1 lecture]
• An approximation technique frequently used in quantum chaos. Examples: approximation for the Hankel function, alternative derivation of the van Vleck propagator (sketch)
5. The trace formula [~3 lectures]
• Gutzwiller's trace formula relates the quantum energy levels of a system to its classical periodic orbits.
• Definitions: level density, Green's function
• Derivation of the trace formula for billiards from the van Vleck propagator.
6. Spectral statistics [~2 lectures]
• The energy levels of chaotic systems have universal statistical properties, in agreement with predictions from random matrix theory.
• Definition of the spectral form factor.
• Brief introduction to random matrix theory.
• How the trace formula can be used to explain universal spectral statistics: sum rule for periodic orbits, diagonal approximation, brief overview of recent research going beyond the diagonal approximation.
There may be minor changes to this syllabus.

## Reading and References

Lecture notes will be provided. (See http://www.maths.bris.ac.uk/~maxsm/qcnotes.pdf for last year's version. The unit does not follow a particular book but useful references are:

• Quantum Chaos: An Introduction, Hans-Juergen Stoeckmann, Cambridge University Press (1999) [an undergraduate textbook on quantum chaos]
• Nonlinear Dynamics and Quantum Chaos, Sandro Wimberger, Springer Verlag (2014) [also an undergraduate textbook]
• Quantum Signatures of Chaos, Fritz Haake, Springer Verlag, 3rd edition (2010) [contains a lot of material on quantum chaos and random matrix theory, parts of chapter 9 and 10 are relevant for this unit]
• Chaos: Classical and Quantum, P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, ChaosBook.org, Niels Bohr Institute, Copenhagen (2010) [a webbook about classical and quantum chaos]

Unit code: MATHM5700
Level of study: M/7
Credit points: 10
Teaching block (weeks): 1 (7-12)
Unit director: Dr Sebastian Muller
Lecturer: Dr Sebastian Muller

## Pre-requisites

MATH35500 Quantum Mechanics or its equivalent in Physics

None

## Methods of teaching

15 lectures with new material. About 3 problem or revision classes. Problem and solution sheets. Lecture notes.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

• 100% from a 1 hour 30 minute exam in May/June

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.