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Physical Review Letters

Chaos and Husimi Distribution Function in Quantum Mechanics

Kin’ya Takahashi, Nobuhiko Saitô

Published Aug 12, 1985• 148 citations
Publisher

Abstract

We study the correspondence between quantum and classical dynamics with Wigner and Husimi distribution functions, respectively. The nature of quantum-mechanical regular and irregular behaviors is discussed with the use of Wigner and Husimi distribution functions. By numerical studies of the chaotic state, the Husimi distribution function is found to be a better representation than the Wigner distribution function, because coarse graining is usually involved implicitly in an observational process using the Wigner distribution function.

DOI
10.1103/physrevlett.55.645

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We study the correspondence between quantum and classical dynamics with Wigner and Husimi distribution functions, respectively. The nature of quantum-mechanical regular and irregular behaviors is discussed with the use of Wigner and Husimi distribution functions. By numerical studies of the chaotic state, the Husimi distribution function is found to be a better representation than the Wigner distribution function, because coarse graining is usually involved implicitly in an observational process using the Wigner distribution function.
The correspondence between classical and quantum mechanics is an important subject for the better understandings of “quantum chaos”. In particular, it is very important to investigate the correspondence between distribution functions in classical mechanics and in phase space representation of quantum mechanics. This is the review of our recent progresses in the study of distribution functions in classical and quantum mechanics, namely distribution functions in classical mechanics and in coarse-grained classical mechanics as well as the Wigner function and the Husimi function. Topics dealt with include formulations of the Wigner representation, the Husimi representation and coarse-grained classical mechanics, and their applications to the analyses of the eigenstates and time developments of the distribution functions.
The quantum dynamics in Husimi representation is studied for the commutation relation, time development of distribution function and eigen distribution function. In the integrable system, the Husimi eigen distribution function is confined almost on the classical torus and can be approximated by Gaussian distribution with the variance (Δξl)2∼ℏ/2(\Delta\xi_{l})^{2}{\sim}\hbar/2(Δξl​)2∼ℏ/2 around the torus. Further, considering the correspondence between quantum and classical mechanics, the Husimi distribution function is a better representation than the Wigner distribution function, because coarse graining is usually involved in observational process.
In this paper, we study the quantum manifestations of classical chaos in phase space using Wigner and Husimi distribution functions. We test the claim that Husimi represents the correspondence better than Wigner does. The results show the claim is valid. We also use a quantum dissipation scheme empirically for classically damped motions often characterized by strange attractors. We believe quantum resemblance to classical distributions can be regarded as signatures of quantum chaos in phase space.
In a recent paper [K. Z\ifmmode \dot{}\else \.{}\fi{}yczkowski, Phys. Rev. A 35, 3546 (1987)] the generalized Husimi distribution was used to investigate the quantum kicked rotator. It was shown that in the classically chaotic region the Husimi distribution displayed a ``rippled irregular shape.'' It was suggested that such behavior could be considered as a qualitative criterion for quantum chaos. In this Comment it is suggested that such behavior is not necessarily associated with quantum chaos. The rippled irregular features may be due to quantum interference effects between superposed states.
Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.
Time development of quantum wave packet is studied in the Hisumi representation for regular and irregular case. The Husimi distribution function spreads on tori or into irregular region for regular or irregular case, respectively and coincides to classical probability distribution function. However by the quantum mechanical interference effects, small packets appear in the process of time development and thus the Husimi distribution function distributes inhomogeneously in contrast to the classical uniform probability distribution function. These small packets are shown to be more stable than classical one.