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Physical Review

Wigner Quantum Density Function in the Classical Limit. Development of a Three-Dimensional WKBJ-Type Solution

J. F. Schipper

Published Aug 25, 1969• 4 citations
Publisher

Abstract

The one-dimensional Wigner quasiprobability function is considered rigorously in the classical limit ($\ensuremath{\hbar}\ensuremath{\rightarrow}0$). With the implicit $\ensuremath{\hbar}$-dependence of the wave function $\ensuremath{\psi}$ ignored in the integral representation, this limit differs from ${|\ensuremath{\psi}(q)|}^{2}\ensuremath{\delta}(p)$ by terms $\frac{o(\ensuremath{\hbar})}{\ensuremath{\hbar}}$. With the implicit $\ensuremath{\hbar}$ dependence of $\ensuremath{\psi}$ accounted for (as it should be), rigorous asymptotic methods are applied, with the results that: (1) in the "classically allowed" region, the limit is an asymptotic exponential function of $q$, multiplied by an energy-conserving $\ensuremath{\delta}$ function; (2) in the "nonclassically allowed" region, this limit is zero. The stationary-phase approach leads nowhere in this regard. A WKBJ-type solution for the radial Schr\"odinger equation is developed, with rigorous error bounds in the different situations. The Wigner function is computed for an ensemble of one-dimensional harmonic oscillators, and an associated phase-space "density matrix" is constructed therefrom. This is compared with the Slater sum for the ensemble---in general, and in the particular cases of high temperature and of $\ensuremath{\hbar}\ensuremath{\rightarrow}0$; where $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, the asymptotic behavior of the Wigner function is explicitly manifest.

DOI
10.1103/physrev.184.1283

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The one-dimensional Wigner quasiprobability function is considered rigorously in the classical limit (\ensuremathℏ\ensuremath→0\ensuremath{\hbar}\ensuremath{\rightarrow}0\ensuremathℏ\ensuremath→0). With the implicit \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ-dependence of the wave function \ensuremathψ\ensuremath{\psi}\ensuremathψ ignored in the integral representation, this limit differs from ∣\ensuremathψ(q)∣2\ensuremathδ(p){|\ensuremath{\psi}(q)|}^{2}\ensuremath{\delta}(p)∣\ensuremathψ(q)∣2\ensuremathδ(p) by terms o(\ensuremathℏ)\ensuremathℏ\frac{o(\ensuremath{\hbar})}{\ensuremath{\hbar}}\ensuremathℏo(\ensuremathℏ)​. With the implicit \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ dependence of \ensuremathψ\ensuremath{\psi}\ensuremathψ accounted for (as it should be), rigorous asymptotic methods are applied, with the results that: (1) in the "classically allowed" region, the limit is an asymptotic exponential function of qqq, multiplied by an energy-conserving \ensuremathδ\ensuremath{\delta}\ensuremathδ function; (2) in the "nonclassically allowed" region, this limit is zero. The stationary-phase approach leads nowhere in this regard. A WKBJ-type solution for the radial Schr"odinger equation is developed, with rigorous error bounds in the different situations. The Wigner function is computed for an ensemble of one-dimensional harmonic oscillators, and an associated phase-space "density matrix" is constructed therefrom. This is compared with the Slater sum for the ensemble---in general, and in the particular cases of high temperature and of \ensuremathℏ\ensuremath→0\ensuremath{\hbar}\ensuremath{\rightarrow}0\ensuremathℏ\ensuremath→0; where \ensuremathℏ\ensuremath→0\ensuremath{\hbar}\ensuremath{\rightarrow}0\ensuremathℏ\ensuremath→0, the asymptotic behavior of the Wigner function is explicitly manifest.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
Abstract When considering quantum systems in phase space, the Wigner function is used as a function of quasi-probability density. Finding the Wigner function is related to the calculation of a Fourier transform of a certain composition of wave functions of the corresponding quantum system. As a rule, knowledge of the Wigner function is not the final goal, and the calculation of mean values of different quantum characteristics of the system is required. An explicit solution of the Schrödinger equation can be obtained only for a narrow class of potentials, therefore, in the majority of cases, numerical methods must be used to find wave functions. As a result, finding the Wigner function is related to the numerical integration of grid wave functions. When considering a one-dimensional system, the calculation of N2 Fourier integrals of the grid wave function is needed. To provide the necessary accuracy for wave functions, corresponding to the higher states of a quantum system, a larger number of grid nodes is needed. The goal of this work was to construct the numerical-analytical method for finding the Wigner function, which allows the number of computational operations to be considerably reduced. The quantum systems with polynomial potentials were considered, for which the Wigner function is represented as a series in some known functions.
Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total ``quasiprobability'' on such a region can be greater than 1 or less than zero.
In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of ℏ can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of ℏ.
Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.