The lectures will be given on demand only if there is a sufficient number of interested students.

Deterministic Chaos (Part of the Schwerpunktmodul: Complex Systems)

The goal of this course is to introduce fundamental concepts of nonlinear dynamical systems. The course covers conservative and dissipative systems, stable and unstable equilibria, bifurcations, routes to chaos, strange attractors, phase space reconstructions and chaos detection.


  • A. J. Lichtenberg, M. J. Liebermann, Regular and Stochastic Motion, Springer 1983.
  • H.-G. Schuster, Deterministic Chaos - An Introduction, Physik-Verlag 1984,
  • E. Ott, Chaos in Dynamical Systems, Cambridge 1993.
  • B. Eckhardt, Chaos, Fischer, 2004

Wave Chaos (Part of the Schwerpunktmodul: Complex Systems)

We will investigate typical concepts used in wave chaotic (quantum chaos). The main investigation will be on the random wave model introduced by Berry [1] in 1978 and in acustic systems by Ebeling [2] in 1979. By this model it is possible to extract predictions for correlations and distribution for different observables quantitatively. Also deviations are discussed, which are mostly due to the effect of the boundaries of the systems. A short discription of other concepts like periodic orbit theory (Gutzwillers trace formula), random matrix theory (RMT) and scattering theory for open systems is planned.

[1] M. V. Berry. Regular and irregular semiclassical wavefunctions. J. Phys. A 10, 2083 (1977).
[2] K. J. Ebeling. Experimental investigation of some statistical properties of monochromatic speckle patterns. Optica Acta 26, 1345 (1979).


  • H.-J. Stöckmann. Quantum Chaos - An Introduction. University Press Cambridge (1999)
  • U. Kuhl, H.-J. Stöckmann, and R. Weaver. Classical wave experiments on chaotic scattering. J. Phys. A 38, 10433 (2005).
  • M. C. Gutzwiller. Chaos in Classical and Quantum Mechanics. Interdisciplinary Applied Mathematics, Vol. 1. Springer New York (1990).
  • P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vattay. Chaos: Classical and Quantum. Niels Bohr Institute Copenhagen (2005). (www.ChaosBook.org).
  • F. Haake. Quantum Signatures of Chaos. Springer Berlin (1991).

Group Theory for Users

It is a common situation that a system one is interested in remains invariant with respect to symmetries, e.g., reflection or rotation. If such symmetries are properly taken into account, problems such as the solution of the Schrödinger equation may be reduced dramatically in complexity. Group theory provides the tools to take advantage of symmetries in a systematic way.

The lecture introduces into the techniques of group theory, e.g. the use of character tables, with particular emphasis on the needs of users. Examples are taken from the quantum mechanics of atoms, molecules and crystals as well as from molecular vibrations. Only basic knowledge of linear algebra is needed.


  • A. D. Boardman, D. E. O'Connor,  P. A. Young, Symmetry and its Applications in Science, McGraw Hill London (1973)
  • M. Tinkham, Group Theory and Quantum mechanics, Dover Publications New York (2003),

Random Matrices (Part of the Schwerpunktmodul: Complex Systems)

Imagine that nearly nothing is known on the details of the interaction of a system one is interested in. This had been the situation nuclear physicists met about 50 years ago. In an approach showing up to be surprisingly successful the Hamiltonian of the system was replaced by a matrix with Gaussian distributed random elements.  Obviously one cannot expect to get any information on an individual system in such a way.  But universal  properties, e.g. the distribution of distances of neighboring eigenvalues,  showed up to be reproduced exceedingly well,  not only in nuclei but in all chaotic systems.

 In the lecture random matrices will be introduced. Techniques will be developed to calculate universal distributions including an introduction into the supersymmetry method to perform Gaussian averages.


  • M. L. Mehta. Random Matrices. Elsevier (2004).
  • F. Haake. Quantum Signatures of Chaos. Springer Berlin (1991).

Microwave Experiments in Complex Systems

All wave systems behave similar with respect to their universal features such as interference, irrespective of the medium and the wave-length. This allows, e.g., to study the quantum mechanics of complex systems, shortly termed "quantum chaos", by means of their equivalents in corresponding microwave setups. Thus it is possible to scale up atomic nuclei of fm size or quantum dots of µm size to the cm size of microwave resonators.  On the other hand it is possible to scale down, e.g., the pacific ocean of Megameter size, to the size of the lab, thus allowing to study the propagation of waves and the formation of caustics in a table size experiment.

 In the lecture an introduction into microwave measuring techniques will be given with examples taken from the subjects listed above.