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Journal of the Physical Society of Japan

Chaos and Time Development of Quantum Wave Packet in Husimi Representation

Kin’ya Takahashi

Published May 15, 1986• 30 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.55.1443

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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.
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 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 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.
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.
In this paper, we investigate the quantum dynamics of wave packets of bound states in the nonlinear Henon–Heiles (HH) system. The time evolution of a wave packet was separated into two processes involving (a) the motion of its first moments and (b) its spreading. On the time scale when wave packet spreading can be neglected, we were able to establish relations between wave packet quantum dynamics expressed in terms of the initial-state population probability P(t) and classical trajectory dynamics. We have demonstrated that the initial states, which are characterized by a periodic time evolution of P(t) over many periods, are related to the five elliptic periodic orbits in the HH system. The decay of P(t) over a few vibrational periods is related to the classical features of quasiperiodic trajectories and can be accounted for in terms of overlap reduction effects. When the center of the initial wave packet is moved further away from the classical periodic orbits, P(t) is characterized by a single initial peak followed by a ’’noisy’’ background. There is no qualitative distinction between the ’’short-time’’ behavior of a wave packet initiated in distant integrable regions and in classical stochastic regions. For long times, wave packet spreading effects set in. Quantum mechanical estimates for the spreading time τs, which depends on the initial conditions, were provided. For wave packets initiated on periodic trajectories, τs is very long, becoming shorter as one moves away from the fixed points. The quantum mechanical long time (t≫τs) behavior of P(t) exhibits small-amplitude noisy time dependence for all initial conditions. This behavior may be described as ’’quantum mechanical stochasticity’’ and is related to the universal feature of irregularity of the spectral resolution of all wave packets in a nonlinear system. From this point of view, all wave packets exhibit ’’quantum stochastic behavior,’’ but there are vast quantitative differences for different initial conditions that are manifested in terms of the values of τs and of the average noisy amplitude. It should be emphasized that this ’’quantum stochasticity’’ is not related to classical stochasticity phenomena.
Relationship between Husimi function ρ H and distribution function ρ CC in coarsegrained classical mechanics is studied where ρ CC is introduced by same manner of coarse-graining in Husimi function. Defining the product of dynamical variables and Poisson brackets in coarse-grained classical mechanics, we reduce the equations for time development of the quantities. Particularly, ρ CC shows good correspondence to ρ H in early stage of time development, but shows poor conformity after quantum interference appeared.