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SYGS: 30 Jahre Gutzwiller Spurformel

SYGS I: HV I

SYGS I.1: Hauptvortrag

Mittwoch, 28. März 2001, 14:30–15:00, S6

The Trace Formula Between Classical and Quantum Mechanics — •Martin Gutzwiller — IBM Research and Yale University

The physics of atoms, molecules, and nuclei is correctly described by quantum mechanics (QM), but our physical intuition comes from classical mechanics (CM). Since the foundations of QM remain difficult to grasp, it is important to work out as far as possible the relations with CM. Until about 30 years ago, however, this task had been carried out successfully only for integrable systems, i.e. only where each degree of freedom could be treated separately in highly symmetrical systems. Even the spectra for simple systems such as the hydrogen-atom in a strong magnetic field (diamagnetic Kepler problem - DKP) or the donor-atom in a semiconductor (anisotropic Kepler problem - AKP) could not be understood, because 2 degrees of freedom are so strongly coupled that perturbation theory is of no help. The trace formula results from a semiclassical approximation of Green’s function. Whereas the left–hand side of the trace formula is a sum over all the resonance denominators in QM for an external disturbance hν, the right-hand side is a sum over all classical periodic orbits where each term consists of an amplitude and a phase = action-integral for the orbit. This mathematical correspondence leads to the explanation of many complicated spectra; it can be generalized for scattering processes, and leads to detailed statistics of the energy-levels where the individual level can no longer be labeled in any sensible manner. In this way the quantization of Bohr and Sommerfeld gets extended at long last to the classically chaotic systems.