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Physical Review A

Variational perturbation theory for the quantum-statistical density matrix

J. Krzyweck

Published Dec 1, 1997• 7 citations
Publisher

Abstract

A variant of variational perturbation theory is developed for a calculation of the quantum-statistical density matrix in thermal equilibrium. For the diagonal elements, which represent the particle distribution, higher-order corrections are calculated, and show excellent agreement with numerical results from Schr"odinger theory.
DOI
10.1103/physreva.56.4410

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We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. For quantum statistical (thermal) field theory, the stochastic variable of the statistical theory is a boundary field configuration. We explore the properties of an effective theory for such boundary configurations and apply it to the computation of the partition function of an interacting one-dimensional quantum-mechanical system at finite temperature. Plots of free energy and specific heat show excellent agreement with more involved semiclassical results. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident end points and includes nonvanishing boundary terms. An appropriately modified expansion into modified Matsubara modes is presented.
The variational approximation to the thermal properties of a quantum-mechanical system proposed by Giachetti and Tognetti [Phys. Rev. Lett. 55, 912 (1985); Phys. Rev. B 33, 7647 (1986)], Feynman and Kleinert [Phys. Rev. A 34, 5080 (1986)], and others is taken as the starting point, and a systematic perturbation series is developed for both the free energy and the particle density. Generalizations of the usual Feynman propagator appear naturally in the formalism. The anharmonic oscillator is treated as an example; the second-order correction to the free energy and the first-order correction to the particle density are evaluated explicitly and shown to lead to significant improvements over the variational solution.
A variational formalism for the determination of the spinless single-particle density matrix $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator X=Y+iZX=Y+iZX=Y+iZ, where $Y=(\frac{1}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})+W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $Z=\ensuremath{-}(\frac{i}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})\ensuremath{-}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j})=\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})$ is a translation operator, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}={\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator WWW employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.
For pt.I see ibid. vol.16, p.3987-4000, (1983). The determinantal formalism for solving the time-dependent Schrodinger equation of the density matrix has been developed, in I, assuming a constant external perturbation. The important case of an harmonic perturbation is presented here. Detailed expressions are derived for diagonal and nondiagonal elements of the response, and the trace conservation theorem is shown to be satisfied at all times and any order. The results are again applied to the simple illustrative quantum system considered in I.