Symmetries in Quantum Mechanics and Statistical Physics

Quantum theory is one of the most successful and most important achievements of modern physics. Quantum theory provides a physical description of microscopic systems. It characterises the dynamics of very small and light-weighted particles like electrons and protons. However, it also serves for a description of the physical and chemical properties of systems made up from such quantum particles (nuclei, atoms, molecules, etc.). Even in statistical physics, which characterises the macroscopic phenomena of many interacting quantum particles, quantum theory is of fundamental importance.

Mathematically a quantum mechanical system is represented by a differential equation, the so-called Schrödinger equation. From the solutions of such equations one may conclude physical properties of the corresponding quantum system. Unfortunately, such solutions are not always easy to find, and, in many cases, analytical solutions cannot be found. Nevertheless, from the symmetries of the Schrödinger equation (e.g. invariance of the quantum system under rotation or translation) one may conclude similar properties for the solutions. Sometimes symmetry considerations are the key for finding such solutions.

In our team we develop solution methods which take the underlying symmetries of the quantum system into consideration. Besides geometric and kinematic symmetries, more recently, also so-called supersymmetries play an important role. With this methods, explicit physical problems in quantum mechanics and statistical physics are investigated.

Literatur
A. Inomata und G. Junker, "Quasi-classical path integral approach to supersymmetric quantum mechanics," Physical Review A 50, 3636-3649 (1994).
G. Junker, "Supersymmetric Methods in Quantum and Statistical Physics," (Springer-Verlag, Berlin, 1996).
G. Junker, H. Leschke und I. Zan, "Explicit thermostatics of Stanley's n-vector model on the harmonic chain by Fourier analysis," Physica A 237, 257-284 (1997).
G. Junker, H. Leschke und H. Karl, "Classical and quantum dynamics of the n-dimensional kicked rotator," Physics Letters A 226, 155-166 (1997).