The influence of quantization on the onset of chaos in Hamiltonian systems: The Kolmogorov entropy interpretation

Abstract

An extension of the concept of Kolmogorov entropy to quantum mechanical systems is given. Using the Kolmogorov entropy as a common basis for discussion, the onset of chaotic motion in classical mechanical and quantum mechanical systems is compared. It is found that if the spectrum of the system is discrete, the Kolmogorov entropy is zero, and therefore that a bounded quantum mechanical system cannot have chaotic motion like that observed for a corresponding classical mechanical system. This analysis leads to the prediction that if the distributions of states are similar for two bounded systems which in the classical limit have, respectively, quasiperiodic and chaotic motion, wave packets for the two systems decay at similar rates as found by Brumer and Shapiro. The relationship between the defined Kolmogorov entropy, previous interpretations of ’’KAMlike’’ onset of chaos in quantum mechanical systems, and the role played by preparation and observation of a system in influencing the intramolecular dynamics, are discussed.