Quantum Statistical Mechanics as an Exact Classical Expansion with Results for Lennard-Jones Helium

TL;DR

This paper develops an exact classical expansion for quantum statistical mechanics, applies it to Lennard-Jones helium, and calculates quantum corrections to thermodynamic properties with simplified computational methods.

Contribution

It introduces a formally exact double perturbation expansion for quantum statistical mechanics based on classical phase space, incorporating non-commutativity and wave function symmetrization effects.

Findings

Quantum corrections for Lennard-Jones helium range from several percent to 1% of classical pressure.

Non-commutativity effects are larger than wave function symmetrization effects.

Corrections are negligible for argon under studied conditions.

Abstract

The quantum states representing classical phase space are given, and these are used to formulate quantum statistical mechanics as a formally exact double perturbation expansion about classical statistical mechanics. One series of quantum contributions arises from the non-commutativity of the position and momentum operators. Although the formulation of the quantum states differs, the present results for separate averages of position operators and of momentum operators agree with Wigner (1932) and Kirkwood (1933). The second series arises from wave function symmetrization, and is given in terms of $l$-particle permutation loops in an infinite order re-summation. The series gives analytically the known exact result for the quantum ideal gas to all orders. The leading correction corrects a correction given by Kirkwood. The first four quantum corrections to the grand potential are calculated for a Lennard-Jones fluid using the hypernetted chain closure. For helium on liquid branch isotherms, the corrections range from several times to 1\% of the total classical pressure, with the effects of non-commutativity being significantly larger in magnitude than those of wave function symmetrization. All corrections are found to be negligible for argon at the densities and temperatures studied. The calculations are computationally trivial as the method avoids having to compute eigenfunctions, eigenvalues, and numerical symmetrization.