Semiclassical Quantization on Fragmented Tori

Abstract

In many cases the method of Einstein-Brillouin-Keller (or EBK quantization on tori) gives excellent semiclassical quantum levels when the classical motion is integrable. Analysis of the primitive semiclassical quantum energy levels suggests a Poisson distribution of nearest neighbor level spacings. Lack of integrability — classical chaos — Is then associated with (i) failure of the EBK method and (ii) level repulsions, and conjectures as to the form of P(S) the normalized level spacing distribution, as S → 0. The expectation that classical chaos leads to robust avoided crossings (strong level repulsion) seems to have been verified by numerical experiment: however, an expected result does not always verify the initial premise. In this lecture it is argued that even in chaotic volumes of phase space nonintegrability sometimes does not completely destroy the underlying time independent manifold structure of classical phase space: Fragments of the invariant tori remain and may be used as a basis for EBK quantization. This is illustrated for the Hénon-Heiles problem, and for the truncated π/4- right triangular rational billiard — both nonintegrable systems. In both cases the underlying quantum level structure follows from integrable approximations to the dynamics, and avoided crossings are easily rationalized via the primitive (as opposed to uniform) quantization used — leading to the conjecture that classical chaos may have little, a per se, to do with the results of currently available numerical experiments.

Chaos … It has also some other significations among the alchemist.

Ephraim Chambers, Cyclopedia (Supplement of 1753)

Visiting Scientist 1982–83; Permanent address: Department of Chemistry, University of Colorado, and Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado, Boulder, Colorado, 80309 USA.