Chaos in classical mechanics

Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York Springer)... 

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Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics Springer-Verlag, New York, 1990. 

MSN.131.1. Prigogine, T. Y. Petrosky, H. H. Hasegawa, and S. Tasaki, Integrability and chaos in classical and quantum mechanics. Chaos, SoUt. Fractals 1, 3-24 (1991). 

This type of energy exchange in an autoionization process may correspond with the behavior of a kicked rotator in classical mechanics, which is known to exhibit chaos. It would be worthwhile to consider an autoionization process of a simple diatomic molecule in its Rydberg states to understand experimentally the essential dynamics of a quantum system, whose classical counterpart exhibits chaos. 

Here, we also include the contributions related to quantum mechanics The chapter by Takami et al. discusses control of quantum chaos using coarsegrained laser fields, and the contribution of Takahashi and Ikeda deals with tunnehng phenomena involving chaos. Both discuss how chaos in classical behavior manifests itself in the quantum counterpart, and what role it will play in reaction dynamics. 

Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

With this section we finish our general survey of chaos in classical and quantum mechanics and turn to a discussion of specific examples of the manifestations of chaos in atomic physics. 

For systems with a continuous spectrum, the concept of chaos is not defined in classical mechanics,66 since at least some trajectories will escape from the interaction region, as t -> oo. One could introduce a definition based on the notion of cantori, which act as barriers in phase space67 and which can serve68 to provide an analogue of a transition-state configuration for systems whose time evolution is constrained. Our own preferred interpretation is based on the discussion of Section III E. The fluctuations in rates are with respect to the zero-time limit of the dynamics. When v = 1, it is not necessary to propagate the system for a finite time in order to be able to predict the rates. The variation in rates are just the inherent fluctuations about the conventional, statistical specific features that... 

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