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Substitute this into Schr_dinger equation. 1_ 2m R _2R _q2 +_1.. _m aq _-qJ = aR a"_-. [pv]=o. a) is a classical Hamilton-Jacobi equation whose solution is S. b) where the velocity v = m 0q , and p = R2 describes conservation of probability. taken alone as a semi-classical no gravity involved. classical zero action of the system that the ratio of h to some with [14]. This limit is also discussed classical perturbation the limit h _ 0 can be approximation in many cases of which there is It really means, however, characteristic approaches Therefore the same dimension in both quantum and theories [15].

54, 1613 (1985); E. G. Gwinn, R. M. Westervelt, Phys. Rev. A. 33, 4143 (1986); D. D'Humieres, M. R. Beasley, B. A. Hubermann, A. Libchaber, Phys. Rev. A. 26, 3483 (1982); J. B. McLaughlin, J. Star. Phys. 24, 375 (1981). [2] This req'aires a little bit of explanation. II. It is too weak to be observable in present time. [3] G. Casati, J. Ford, I. Guarneri, F. Vivaldi, Phys. Rev. A. 34, 1413 (1986); see also for classical mapping equations corresponding to ours, B. V. Chirikov, Phys. Rep. 52, 263 (1979); see also Quantum Chaos edited by H.

H. Jensen, Phys. Rev. A. 45, 2686 (1992). [16] D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952); Nature, 315, 294 0985). [17] I follow here J. J. Sakurai's argument. See his quantum text book, Modern Quantum Mechanics (Benjamin Cummings, Menlo Park, CA, 1985). [18] For a comprehensive argument example, D. H. W. about taking a classical limit, see for Science, 253, 626 (1991) Noid, M. L. Koszykowski, R. A. Marcus, Annu. Rev. Phys. Chem. 32, 267 (1981); W. P. Reinhardt, Chaotic Behaviors in Quantum Systems: Theory and Applications, edited by G.

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