Dynamical peculiarities of the nonlinear quasiclassical systems

The quantum-classical correspondence for dynamics of the nonlinear classically chaotic systems is analysed. The problem of quantum chaos consists of two parts: the quasiclassical quantisation of the chaotic systems and attempts to understand the classical chaos in terms of quantum mechanics. The first question has been partially solved by the Gutzwiller semiclassical trace formula for the eigenvalues of chaotic systems, while the classical chaos may be derived from quantum equations only introducing the decoherence process due to interaction with system's environment or intermediate frequent measurement. We may conclude that continuously observable quasiclassical system evolves essentially classically-like. Quasiclassical systems are systems with high quantum numbers and they may be described quantum mechanically using the Bohr-Sommerfeld or WKB quantization method. The quasiclassical (semiclassical) methods are built on the classical trajectories. If a classical mechanical system can be represented as multiperiodic, than it is equivalent to as many independent systems as many degrees of freedom, or as many constants of the motion the system has. Einstein, quoting Poincare, pointed out that such a separation was not possible in the three-body problem and for, what he called, " type B " (chaotic) classical motion. However this statement of the Einstein was completely ignored for many years. Classical trajectories of the nonlinear systems are extremely complex, even chaotic. Namely the classical dynamical chaos has prevented the broad application of semiclassi-cal ideas and techniques. After much effort Bohr, Krammers, Born and Heisenberg failed in their attempt to quantize the helium atom. They neither understood that classical helium atom is chaotic nor understood dynamical chaos in general. Only in 1970 Gutzwiller derived his semiclassical trace formula for the eigenvalues of chaotic systems (see, [1] and references therein). Quasiclassical theory is an approximation of the quantum mechanics and works very well in a great variety of systems and large range of parameters, e.g. results to the very precise approach for the matrix elements [2]. That is way the semi-classical theory, being really worthwhile for the better understanding of many atomic, molecular and mesoscopic processes has been named " postmodern quantum mechanics " [3]. However it does not solve all problems of the quantum-classical correspondence for the dynamics of the nonlinear systems. 1 In general, the quantum and classical realms are related by the correspondence principle: physical characteristics of the highly exited quantum systems with large quantum numbers are close to those of its classical counterpart. However, recent research …