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Quantum Chaos (MATH M5700)

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Administrative Information

  1. Unit number and title: MATH M5700 Quantum Chaos
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2005-06, with changes from 2010-11 & 2011-12
  6. Unit Organiser: Sebastian Muller
  7. Lecturer: Dr Sebastian Müller
  8. Teaching block: 2
  9. Prerequisites: MATH35500 Quantum Mechanics or equivalent for Physics students

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 reflected 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.

 

Teaching Methods

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

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 (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. 

Assessment Methods

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.

Award of Credit Points

Credit points are gained by:

  • either passing the unit (i.e. gaining a mark of 50 or over);
  • or getting an assessment mark of 30 or over and also handing in satisfactory attempts of 75% of homework assignments.

Transferable Skills

  • Clear, logical thinking.
  • Problem solving techniques.
  • Assimilation and use of complex and novel ideas.

Texts

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]

Syllabus

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.