Semiclassical ergodic properties of the Henon–Heiles system

This paper investigates the applicability of the concepts of classical ergodicity and the results of semiclassical ergodic theory to quantum mechanical systems that are away from the classical limit and that do not necessarily become ergodic as ℏ → 0. To carry out this study, a new quantity, called the ‘‘constancy’’ FA of classical property A, is introduced. This constancy is defined in terms of the autocorrelation function of A and its statistical equilibrium value. It measures, on an absolute scale, the extent to which the dynamics of A behave statistically. The FA for a set of quadratically integrable properties A can be used to define the ‘‘degree of ergodicity’’ of a classical system with respect to this set. This analysis motivates introduction of the quantum analog of FA as a means of judging the applicability of classical ergodicity concepts to quantum mechanical systems. To illustrate these ideas, calculations for the Henon–Heiles system are carried out which compare the quantum and classical analogs of the constancies and of the ‘‘almost microcanonical’’ autocorrelation functions from which they are formed. The results indicate that the quantum and classical systems exhibit similar forms of partial ergodicity at high energy. This conclusion supports the approach introduced here for identifying the quantum implications of classical ergodicity. As a consequence of the good quantum-classical agreement, the partially ergodic nature of the classical behavior is reflected in the distribution of certain quantum-mechanical matrix elements. At lower energy, the applicability of classical ergodicity concepts to this system is more limited due to differences in the quantum and classical dynamics. The kinds of quantum-classical discrepancies that limit the implications of classical ergodicity for quantum systems are identified.