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Journal of Physics A Mathematical and General

Semiclassical densities in phase space: Wigner's distribution function for thermodynamic equilibrium

H. J. Korsch

Published Jun 1, 1979• 7 citations
Publisher

Abstract

The quantum-mechanical joint (quasi-) density in phase space (the Wigner distribution) for thermodynamic equilibrium is studied in the semiclassical limit h(cross) to 0. The approximation is of an asymptotic (WKB) type and nonperturbative in character. The equilibrium phase-space distribution can be expressed in terms of classical paths satisfying certain well defined conditions. These paths are complex-valued, i.e. classically forbidden. The resulting semiclassical expression for the equilibrium Wigner phase-space density agrees with the exact quantum result for the analytical solvable case of harmonic potentials for all temperatures.

DOI
10.1088/0305-4470/12/6/010

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Journal of Physics Conference Series•Nov 1, 2016•1 citation
The quantum-mechanical joint (quasi-) density in phase space (the Wigner distribution) for thermodynamic equilibrium is studied in the semiclassical limit h(cross) to 0. The approximation is of an asymptotic (WKB) type and nonperturbative in character. The equilibrium phase-space distribution can be expressed in terms of classical paths satisfying certain well defined conditions. These paths are complex-valued, i.e. classically forbidden. The resulting semiclassical expression for the equilibrium Wigner phase-space density agrees with the exact quantum result for the analytical solvable case of harmonic potentials for all temperatures.
The phase space formulation of quantum statistical mechanics using the Feynman path centroid density offers an alternative perspective to the standard Wigner prescription for the classical-like evaluation of equilibrium and/or dynamical quantities of statistical systems. The use of this formulation has been implicit in recent work on quantum rate theories, for example, in which the centroid density distribution replaces the classical Boltzmann distribution. In order to further understand the approximations involved in this and similar transcriptions, the present work elaborates and clarifies the issue of operator ordering in a rigorous centroid-based formulation. In particular, through the use of the Weyl correspondence, a precise definition of the centroid symbol of operators and their products is presented. Though we fall short of finding the algebraic structure tantamount to that found in the Weyl symbols—of which the Wigner distribution is an example— the resulting expressions have internal consistency and are amenable to approximate evaluation through cumulant expansions.
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 qqq, 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.
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.
Semiclassical approximations to quantum dynamics are almost as old as quantum mechanics itself. In the approach pioneered by Wigner, the evolution of his quasiprobability density function on phase space is expressed as an asymptotic series in increasing powers of Planck's constant, with the classical Liouvillean evolution as leading term. Successive semiclassical approximations to quantum dynamics are defined by successive terms in the series. We consider a complementary approach, which explores the quantum-clssical interface from the other direction. Classical dynamics is formulated in Hilbert space, with the Groenewold quasidensity operator as the image of the Liouville density on phase space. The evolution of the Groenewold operator is then expressed as an asymptotic series in increasing powers of Planck's constant. Successive semiquantum approximations to classical dynamics are defined by successive terms in this series, with the familiar quantum evolution as leading term.
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.