Abstract
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.
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References
Berry, M.V. (1981): Ann. Phys. 131 163.
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).
Berry, M.V. and Tabor, M. (1977a): J. Phys. A: Math. Gen. 10 371.
Berry, M.V. and Tabor, M. (1977b): Proc. Roy. Soc. Lond. A356 375.
Berry, M.V. and Wilkenson, M. (1984): Proc. Roy. Soc. Lond. A392 15.
Bohigas, O., Giannoni, M.J., and Schmidt, C. (1984): Phys. Rev. Lett. 52 1.
Brody, T.A. (1974): Lett. Nuovo Cimento 7 482.
Brody, T.A., Flores, J., French J.B., Mello, P.A., Pandey, A., and Wong, S.S.M. (1981): Rev. Mod. Phys. 53 385.
Brookes, B.C. and Dick, W.F.L. (1969): Introduction to Statistical Methods (Heinemann, London).
Camarda, H.S. and Georgopulos, P.D. (1983): Phys. Rev. Lett 50 492.
Casati, G., Vals-Gris, F., and Guarnieri, I. (1980): Lett. Nuovo Cimento 28 279.
Casati, G., Chirikov, B.V., and Guarnieri, I. (1985): Phys. Rev. Lett. 54 1350.
Cheng, Z. and Lebowitz, J.L. (1991): “Statistics of Energy Levels in an Integrable Quantum System” (Preprint: Rutgers University)
Feingold, M. and Peres, A. (1986): Phys. Rev. A34 591.
Gaspard, P., Rice, S.A., and Nakamura, K. (1989): Phys. Rev. Lett. 63 930.
Gibbons, J. and Hermsen, T. (1984): Physica Dl1 337.
Goggin, M.F. and Milonni, P.W. (1988): Phys. Rev. A37 796.
Haake, F. (1990): Quantum Signatures of Chaos (Springer-Verlag, Berlin).
Haake, F., Kus, M., Scharf, R. (1987a): Z. Physik B65 381.
Haake, F., Kus, M., Scharf, R. (1987b): in Fundamentals of Quantum Optics, II, Lecture Notes in Physics. Edited by F. Ehlotzky (Springer-Verlag, Berlin).
Hacken, G. Werbin, R. and Rainwater, J. (1978): Phys. Rev. C17 43.
Haller, E., Koppel, H., and Cederbaum, L.S. (1983): Chem. Phys. Lett. 101 215.
Haq, R.U., Pandey, A. and Bohigas, O. (1982): Phys. Rev. Lett. 48 1086.
Knox, R.S. and Gold, A. (1964): Symmetry in the Solid State (W.A. Benjamin, New York).
Kus, M., Haake, F., Scharf, R. (1987): Z. Physik B66 129.
Liou, H.I., Camarda, H.S., Wynchank, S., Slagowitz, M., Hacken, G., Rahn, F., and Rainwater, J. (1972): Phys. Rev. C5 974.
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.)
McDonald, S.W. and Kaufman, A.N. (1979): Phys. Rev. Lett. 42 1189.
Meyer, S.L. (1975): Data Analysis for Scientists and Engineers (J. Wiley and Sons, Inc. New York).
Nakamura, K. and Lakshmann, M. (1987): Phys. Rev. Lett. 57 1661.
Nakamura, K. and Mikeska, H.J. (1987): Phys. Rev. A35 5294.
Pechukas, P. (1983): Phys. Rev. Lett. 51 943.
Percival, I.C. (1973): J. Phys. B6 L229.
Peres, A. (1991): in Quantum Chaos. Edited by H.A. Cerdeira, R. Ramaswamy, M.C. Gutzwiller, and G. Casati (World Scientific, Singapore).
Prochnow, N.H., Newson, H.W., Bilpuch, E.G.,and Mitchell, G.E. (1972): Nucl.Phys. A194 353.
Scharf, R., Dietz, B., Kus, M., Haake, F., and Berry, M.V. (1988): Europhysics Lett. 5 383.
Seligman, T.H. and Verbaarshot, J.J.M. (1986): Phys. Rev. Lett. 56 2767.
Seligman, T.H., Verbaarshot, J.J.M., and Zirnbauer, M.R. (1984): Phys. Rev. Lett. 53 215.
Smalley, R.E., Wharton, L., and Levy, D.H. (1983): J. Chem. Phys. 63 4977.
Srivastava, N., Kaufman, C., Muller, G. (1990a): J. Applied Phys. 67 5627.
Stockmann, H.-J. and Stein, J. (1990): Phys. Rev. Lett. 64 2215.
Teller, E. (1937): J. Phys. Chem. 41 109.
Terasaka, T. and Matsushita, T. (1985): Phys. Rev. A32 538.
von Neumann, J. and Wigner, E.P. (1929): Phys. Z. 30 467. (Translated into english in [Knox and Gold 1964].)
Yukawa, T. (1985): Phys. Rev. Lett. 54 1883.
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Reichl, L.E. (1992). Observed Spectra. In: The Transition to Chaos. Institute for Nonlinear Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4352-4_7
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