In classical systems we have a clear picture of the mechanism by which constants of the motion are destroyed, because constants of the motion constrain phase space trajectories to surfaces of lower dimension (KAM surfaces). When these surfaces are destroyed, trajectories are free to wander in a chaotic manner throughout regions of higher dimension. This whole process is easily seen numerically in Poincare surfaces of section. In quantum systems, we cannot define phase space trajectories because we cannot simultaneously specify both the momentum and position of a particle with arbitrarily high precision due to the uncertainty principle. However, what has proven to be extremely useful is a study of the statistical properties of quantum systems whose classical counterparts undergo a transition from regular (dominated by KAM surfaces) to chaotic behavior. Indeed, Percival [Percival 1973] suggested that this might be so, at least in the semiclassical limit, since when a constant of the motion is destroyed so is the quantum number that one might assign to it.
KeywordsSpectral Sequence Wigner Distribution Spectral Statistic Spectral Spacing Neighbor Spacing
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- Berry, M.V. (1986): in Quantum Chaos and Statistical Nuclear Physics (Lecture Notes in Physics, Vol. 263) edited by T.H. Seligman and H. Nishioka (Springer-Verlag, Berlin).Google Scholar
- Brookes, B.C. and Dick, W.F.L. (1969): Introduction to Statistical Methods (Heinemann, London).Google Scholar
- Cheng, Z. and Lebowitz, J.L. (1991): “Statistics of Energy Levels in an Integrable Quantum System” (Preprint: Rutgers University)Google Scholar
- Haake, F. (1990): Quantum Signatures of Chaos (Springer-Verlag, Berlin).Google Scholar
- Haake, F., Kus, M., Scharf, R. (1987b): in Fundamentals of Quantum Optics, II, Lecture Notes in Physics. Edited by F. Ehlotzky (Springer-Verlag, Berlin).Google Scholar
- Martin, W.C., Zalubas, R., and Hagan, L. (1978): Atomic Energy Levels — The Rare Earth Elements U.S. National Bureau of Standards, National Standards Reference Data Series — 60 (U.S. GPO, Washington, D.C.)Google Scholar
- Meyer, S.L. (1975): Data Analysis for Scientists and Engineers (J. Wiley and Sons, Inc. New York).Google Scholar
- Peres, A. (1991): in Quantum Chaos. Edited by H.A. Cerdeira, R. Ramaswamy, M.C. Gutzwiller, and G. Casati (World Scientific, Singapore).Google Scholar