Classical and Quantum Chaos in Atomic Systems

Abstract

Atomic systems played a major role in the birth and growth of quantum mechanics. One central idea was to relate the well-known classical motion of the electron of a hydrogen atom--an ellipsis around the nucleus--to the experimentally observed quantization of the energy levels. This is the aim of the Bohr and Bohr-Sommerfeld models. These simple semiclassical models were unable to make any reliable prediction on the energy spectrum of the next simplest atom, helium. Because of the great success of quantum mechanics, the problem of correspondence between the classical and the quantal dynamics has not received much attention in the last 60 years. The fundamental question is (Gutzwiller, 1990). How can classical mechanics be understood as a limiting case within quantum mechanics? For systems with time-independent one-dimensional dynamics like the harmonic oscillator and the hydrogen atom, the correspondence is well understood. The restriction to such simple cases creates the erroneous impression that the classical behavior of simple systems is entirely comprehensible and easily described. During the last 20 years it has been recognized that this in not true and that a complex behavior can be obtained from simple equations of motion. This usually happens when the motion is chaotic, thatmore » is, unpredictable on a long time scale although perfectly deterministic (Henon, 1983). A major problem is that of understanding how the regular or chaotic behavior of the classical system is manifest in its quantum properties, especially in the semiclassical limit. 53 refs., 15 figs., 1 tab.« less