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Europhysics Letters (EPL)

A Remark on the Problem of Quantum-Classical Correspondence in the Case of Chaotic Dynamics

Andrey R. Kolovsky

Published Jul 10, 1994• 26 citations
Publisher

Abstract

The paper discusses the problem of correspondence between the classical and quantum approaches to the description of chaotic dynamics of a mesoscopic particle. The condition for the equivalence of both approaches is given.
DOI
10.1209/0295-5075/27/2/001

Related papers

Comment on “A Remark on the Problem of Quantum-Classical Correspondence in the Case of Chaotic Dynamics”

Juan Cesar Flores

In the paper by Kolovsky ( Europhys. Lett. , 27 (1994) 79) an equation for a mesoscopic particle, with internal structure, was proposed. Nevertheless, that equation is in confrontation with the stability of such a mesoscopic particle. This instability is independent of the chaotic character of the corresponding classical system.

Europhysics Letters (EPL)•Mar 10, 1995•6 citations

Comment on “A Remark on the Problem of Quantum-Classical Correspondence in the Case of Chaotic Dynamics”

A. Pérez Ponce

In a very interesting article written by A. R. Kolovsky, a first-order multipole evolution equation for a mesoscopic particle with internal structure was proposed. In some physical cases the first-order approximation breaks down and it is the second order which must be considered. In this way, we reconsider the case of a mesoscopic particle constituted by identical particles.

Europhysics Letters (EPL)•Oct 1, 1995•3 citations

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arXiv:quant-ph/9610041•Oct 28, 1996•0 citations
The quantum dynamics of a classically chaotic model are studied in the approach to the macroscopic limit. The quantum predictions are compared and contrasted with the classical predictions of both Newtonian and Liouville mechanics. The time-domain scaling of the optimal quantum-classical correspondence is analyzed in detail in the case of both classical theories. In both cases the correspondence for observable quantities is shown to break down on a time-scale that increases very slowly (logarithmically) with increasing system size. In the case of quantum-Liouville correspondence such a short time-scale does not imply a breakdown of correspondence since the largest quantum-Liouville differences reached on this time-scale decrease rapidly (as an inverse power) with increasing system size. Therefore the statistical properties of chaotic dynamics are, as expected, well described by quantum theory in the macroscopic limit. In contrast, the largest quantum-Newtonian differences reached on the log time-scale actually increase in proportion to the system size. Since the invariance properties of the Hamiltonian impose functional constraints on the time-varying chaotic coordinates, it is possible to show, moreover, that if the quantum predictions are believed to describe the coordinates of individual chaotic systems, then they also predict macroscopic violations of any kinematic or dynamic constants of the motion. These results for chaotic systems indicate that a valid description of the time-varying properties of individual macroscopic bodies is not available within the standard interpretive framework of quantum theory.
Mesoscopic devices, with system sizes in the range of several to several dozens wavelengths, represent paradigmatic model systems for the observation of quantum chaotic behaviour based on semiclassical concepts. Those electronic and photonic billiard cavities are small enough for interference effects not to be ignored. Nonetheless, the classical ray or particle tracing picture can often provide a substantial understanding of the dynamics of the system along the lines of classical-quantum, or ray-wave correspondence. This well-established principle turns out to be particularly useful when applied not only in real space, but by extending it to phase space such that both location and momentum information can contribute to a deeper and more comprehensive understanding of the dynamical behaviour.
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.