# From classical to quantum mechanics with hard chaos

## Abstract

Most of the interesting dynamical systems in physics and chemistry are characterized by an intimate mixture of two extremes: integrability on one hand, and "hard chaos" on the other. The first of these is well understood, particularly as one tries to connect classical and quantum mechanics. The second, however, has been the exclusive preserve of the mathematicians and suffers, moreoever, from a dearth of good physical examples, although it is probably more frequent in nature than its integrable counterpart. The energy surface in the phase space of a harshly chaotic system foliates into two sets of smooth manifolds, the stable and the unstable ones, each of dimension equal to the number of degrees of freedom. A classical trajectory is the intersection of a stable and an unstable manifold. Its physical properties can be described by a simple coding scheme that gives a unique account of its whole history. As an example, the anisotropic Kepler problem (AKP), isotropic Coulomb potential with anisotropic mass tensor, is discussed in some detail, because it allows the two sets of manifolds to be constructed by varying only a single parameter and because its code is the simplest imaginable, consisting of all binary sequences. The transition to quantum mechanics, using Green's function and its trace, leads to expressions that resemble very closely the grand-canonical partition function of a one-dimensional Ising chain. The energy levels are the zeros in the complex (1/kT) plane and should be located on the imaginary axis. This analogy is directly related to the underlying hard chaos and is followed up by studying the locations of these zeros for various coupling parameters in the Ising chain. It appears that some further restrictions have to be invoked, in order to guarantee a reasonable transition from classical to quantum mechanics in the case of hard chaos. © 1988 American Chemical Society.