Quantum mechanics and quantum chaos

Marko Robnik

N. G. van Kampen

James T. Cushing

The ubiquity of chaos in classical mechanics (CM), as opposed to the situation in standard quantum mechanics (QM), might be taken as speaking against QM being the fundamental theory of physical phenomena. Bohmian mechanics (BM), as a formulation of quantum theory, may clarify both the existence of chaos in the quantum domain and the nature of the classical limit. Two interesting possibilities are (i) that CM and classical chaos are included in and underwritten by quantum mechanics (BM) or (ii) that BM and CM simply possess a common region of (noninclusive) overlap. In the latter case, neither CM nor QM alone would be sufficient, even in principle, to account for all of the physical phenomena we encounter. In this talk I shall summarize and discuss the implications of some recent work on chaos and on the classical limit within the framework of BM.

Giulio Casati

Mason Alexander Porter

Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.

C. Kratochvíl, Petr Švéda