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

Unit code | MATHM5700 |

Credit points | 10 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 1B (weeks 7 - 12) |

Unit director | Dr. Muller |

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.

**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 reď¬‚ected 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.

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.

Mechanics 2/23 may become a prerequisite from 2021/22.

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.

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

100% Timed, open-book 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.

**Recommended**

- P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner and G. Vattay,
*Chaos: Classical and Quantum*Niels Bohr Institute, Copenhagen, 2010 - ChaosBook.org - Fritz Haake,
*Quantum Signatures of Chaos*, Springer Verlag, 3rd edition, 2010 - Hans-Juergen Stoeckmann,
*Quantum Chaos: An Introduction*, Cambridge University Press, 1999 - Sandro Wimberger,
*Nonlinear Dynamics and Quantum Chaos*, Springer Verlag, 2014