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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: 08/09
  5. First Given in this form: 2005-06
  6. Unit Organiser: Francesco Mezzadri
  7. Lecturer: Dr F Mezzadri
  8. Teaching block: 2
  9. Prerequisites: 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

Quantum Chaos studies the mathematical and physical properties that in quantum systems are signatures of the chaotic nature of the underlying classical mechanics. At microscopic length scales, the chaotic dynamics of the corresponding classical system manifests himself 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 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 formula 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.

Relation to Other Units

The unit requires basic knowledge of quantum mechanics, which is a prerequisite. Some ideas discussed are related to topics presented in the level 3 unit Random Matrix Theory.

Teaching Methods

Lecture, problem classes, homework excercises. Notes will be made available to the students.

Learning Objectives

After completing the unit successfully you will be able to:

  • Define and illustrate the notions of Random Matrix theory conjecture for spectral correlations and Quantum Ergodicity for quantum mechanical systems.
  • Recall and derive the main mathematical and physical properties of few examples of quantum maps.
  • Apply such properties 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;
  • 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

There is no recommended text but two useful references are:

  • Fritz Haake. Quantum Signatures of Chaos, Springer Verlag, 2000.
  • Ozorio de Almeida. Hamiltonian Systems: Chaos and Quantization, Cambridge University Press, 1990

Syllabus

  • Introduction: motivations and examples. 1 lectures.
  • Ergodic properties of classical mechanics. 3 lectures.
    1. Review of Hamiltonian mechanics.
    2. Ergodicity, mixing and hyperbolic dynamics.
    3. Hannay and Ozorio de Almeida sum rule for maps.
  • Maps and their quantization. 3 lectures
    1. Torus kinematic and quantization of the linear automorphisms of the two-torus (cat maps).
    2. Nonlinear perturbations of the cat maps.
  • The random matrix theory conjecture and semiclassical dynamics. 5 lectures.
    1. The RMT conjecture and spectral statistics, the Berry-Tabor conjecture.
    2. The trace formula for maps.
    3. The diagonal approximation and the form factor.
  • Quantum ergodicity and the random wave model. 3 lectures.
    1. Egorov theorem, quantum ergodicity, quantum unique ergodicity and scars; the example of the quantized cat maps.
    2. The random wave model