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. An important example are billiards. These will play an important role in the course as they are amenable to a clean mathematical treatment as well as relevant for applications. However there are many more examples. On a billiard table, a particle moves on a straight line and is reflected at the boundary. If the shape of the billiard table is irregular the classical motion of the billiard ball depends sensitively on the initial condition. Moreover long trajectories tend to fill the billiard table uniformly and all directions are equally likely; this property is called ergodicity.

If a system is classically chaotic this has important consequences for its quantum mechanical behaviour. In particular energy eigenfunctions corresponding to large energy levels display a phenomenon called quantum ergodicity. Roughly speaking, for these energy eigenfunctions the probability to find a particle in a (sufficiently large) part of the billiard becomes the same for all parts of the same area. Moreover the energy levels display a universal statistical behaviour for all chaotic systems. They tend to ‘repel’ each other i.e. it is extremely rare to have two energy levels very close to each other. Many other quantum mechanical properties of chaotic systems display similar universal behaviour.

The course starts with an introduction into the classical properties of chaotic systems as well as quantum ergodicity. We will then develop methods to relate the quantum mechanical features of a system to its classical dynamics, by approximating quantum mechanical properties in terms of sums over classical trajectories. In general semiclassical approximations like this are very helpful to understand the behaviour of systems that are small enough such that quantum phenomena are important, but large enough such that classical mechanics provides useful insight. This semiclassical approach will be used to study quantum mechanical time evolution as well as quantum mechanical energy levels. Central results will be the so-called van Vleck propagator as well as the Gutzwiller trace formula. The links between classical and quantum mechanics thus established will then be used to investigate the statistics of energy levels for chaotic systems, and present elements of the theoretical understanding of the origin of level repulsion. We will also discuss (in a self-contained way) the connection of this approach to Random Matrix Theory.

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.


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)
Lecturer: Dr Sebastian Müller


MATH35500 Quantum Mechanics or its equivalent in Physics



Methods of teaching

Lectures and 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

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.

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