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Physical Review A

Wigner-Kirkwood expansions

Y. Fujiwara, T. A. Osborn, S. F. J. Wilk

Published Jan 1, 1982• 100 citations
Publisher

Abstract

This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function $F$. For an $N$-body quantum system with Hamiltonian $H$ and inverse temperature $\ensuremath{\beta}$, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: $F=\frac{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}H}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}{H}_{0}}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}$. Expansion of $\mathrm{ln}F$ in Planck's constant $h$ leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable $\ensuremath{\beta}$ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.

DOI
10.1103/physreva.25.14

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This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function FFF. For an NNN-body quantum system with Hamiltonian HHH and inverse temperature \ensuremathβ\ensuremath{\beta}\ensuremathβ, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: F=〈x\ensuremath→∣e\ensuremath−\ensuremathβH∣x\ensuremath→\ensuremath′〉〈x\ensuremath→∣e\ensuremath−\ensuremathβH0∣x\ensuremath→\ensuremath′〉F=\frac{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}H}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}{H}_{0}}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}F=〈x\ensuremath→∣e\ensuremath−\ensuremathβH0​∣x\ensuremath→\ensuremath′〉〈x\ensuremath→∣e\ensuremath−\ensuremathβH∣x\ensuremath→\ensuremath′〉​. Expansion of lnF\mathrm{ln}FlnF in Planck's constant hhh leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable \ensuremathβ\ensuremath{\beta}\ensuremathβ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.
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This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator exp(+sH)\mathrm{exp}(+sH)exp(+sH), where HHH is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary ccc-number integral of a ccc-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian, Heff{H}_{\mathrm{eff}}Heff​, which is a function of the parameter sss and the ccc-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function ZN{Z}_{N}ZN​ in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for Heff{H}_{\mathrm{eff}}Heff​ depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying exp(sHeff)\mathrm{exp}(s{H}_{\mathrm{eff}})exp(sHeff​). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for Heff{H}_{\mathrm{eff}}Heff​. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
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