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 |
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.
At the end of the unit the student should:
Transferable Skills:
15 lectures with new material. About 3 problem or revision classes. Lecture notes.
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.
Lecture notes will be provided. The unit does not follow a particular book but useful references are: