Unit name | Quantum Chaos |
---|---|

Unit code | MATHM5700 |

Credit points | 10 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 2C (weeks 13 - 18) |

Unit director | Professor. Jon Keating |

Open unit status | Not open |

Pre-requisites |
MATH35500 Quantum Mechanics or its equivalent in Physics |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

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.

General Description of the Unit

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, and the semiclassical approximations in this unit are examples for asymptotic approximations studied in the Asymptotics unit. All relevant material connected to these units will be introduced in a self-contained way.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

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.

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

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

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]
- 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]