Unit information: Quantum Chaos in 2013/14

Unit name	Quantum Chaos
Unit code	MATHM5700
Credit points	10
Level of study	M/7
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Muller
Open unit status	Not open
Pre-requisites	Mechanics 2 or Mechanics 23 and Quantum Mechanics or equivalent for Physics students.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Quantum Chaos studies the mathematical and physical properties that in quantum systems are signatures of the chaotic nature of the underlying classical mechanics. At miscropscopic length scales, the chaotic dynamics of the corresponding classical system manifests himself in the behaviour of the eignfunctions 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 ergodic 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, like the Gutzwiller trace formla and the random wave model. Most of the topics will be presented within the framework of systems with a discrete time dynamics (quantum maps), as they often allow a thorough mathematical treatment. 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 unique ergodicity.

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.

Intended Learning Outcomes

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 (Poisson's summation formula, semiclassical 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.

Teaching Information

15 lectures with new material. About 3 problem or revision classes. Lecture notes.

Assessment Information

The assessment mark for Quantum Chaos is calculated from a 1½-hour written examination in May/June consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted in this examination.

The examination questions will assess your knowledge and comprehension of the material taught during the course and your ability to apply the mathematical techniques learnt to typical problems in Quantum Chaos.

Reading and References

Lecture notes will be provided. 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]