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Journal of the Physical Society of Japan

Similarity and Essential Defference between Coarse-Grained Classical and Quantum Mechanics

Kin’ya Takahashi

Published Oct 15, 1989• 2 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.58.3514

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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.
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.
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.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.
By using the formulation of quantum mechanics with classical pictures, the motion of a quantum is analysed for linear harmonic oscillator. On this occasion it is postulated that the quantum mechanical momentum of the quantum is approximately equal to the corresponding classical mechanical momentum in the region of validity of the coordinates. According to this analysis it is shown that the quantum mechanical motion well corresponds to the classical motion, and that the wave equation determines the associated wave along the whole orbit of the quantum in the configuration space and the uniqueness of the wave throughout the orbit plays an important role in quantum mechanics. In addition to the above, an attempt is made to make clear the well-known fact that classical mechanics with the quantization gives approximately the same results as quantum mechanics.