Path-integral approach to the Wigner-Kirkwood expansion

TL;DR

This paper introduces a new path-integral-based method for the Wigner-Kirkwood expansion, enabling efficient calculation of high-temperature quantum thermodynamic properties, demonstrated on the anharmonic oscillator.

Contribution

It presents a novel functional representation of the Wigner-Kirkwood expansion derived from the Feynman-Kac formula, improving analytic computation of higher-order coefficients.

Findings

Efficient generation of higher-order expansion coefficients

Application to thermodynamic functions of the anharmonic oscillator

Discussion on generalization and comparison with world-line formulation

Abstract

We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a new functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in generating analytic form of the higher-order expansion coefficients. To put some flesh on the bare bones we apply the expansion to obtain basic thermodynamic functions of the one-dimensional anharmonic oscillator. Further salient issues, such as generalization to the Bloch density matrix and comparison with the more customary world-line formulation are discussed.