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Progress of Theoretical Physics Supplement

Distribution Functions in Classical and Quantum Mechanics

Kin’ya Takahashi

Published Jan 1, 1989• 229 citations
Publisher

Abstract

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.

DOI
10.1143/ptps.98.109

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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.
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.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
This paper deals with how a classical mechanics relates to the quantum mechanics, comparing the stationary distribution function on a constant energy in a coarse-grained classical system with the Husimi function ρ H of the eigenstate of quantum system. In the domain of a torus with one degree of freedom, these two functions are quite similar to each other. On the other hand, around a hyperbolic fixed-point the functions satisfy the equations of the similar type. However the solutions admissible for classical and quantum distribution functions have very different forms. In this paper we discuss the necessity for the problem of the correspondence between classical and quantum mechanics to investigate not only the similarity of the equations but also the geometrical structure of the solutions.
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.
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.