Wigner-Kirkwood expansion for semi-infinite quantum fluids

TL;DR

This paper extends the Wigner-Kirkwood semiclassical expansion to semi-infinite quantum fluids near a hard wall boundary, revealing new position-dependent and non-analytic terms, and provides explicit quantum corrections for Coulomb fluids.

Contribution

It generalizes the Wigner-Kirkwood expansion to semi-infinite quantum fluids with boundaries, including position-dependent and non-analytic terms, and derives explicit quantum corrections for Coulomb systems.

Findings

Derived position-dependent Boltzmann density terms near a boundary.

Identified non-analytic behavior in the semiclassical expansion due to boundary effects.

Provided explicit quantum correction formulas for Coulomb fluids.

Abstract

For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength $\lambda$. This result permits one to generate an analogous $\lambda$-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type $\exp(-2x^2/\lambda^2)$ ($x$ denotes the distance from the wall boundary) which are non-analytic in $\lambda$. Under some condition, the analyticity in $\lambda$ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of $\lambda$, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the $\nu$-dimensional (integer $\nu\ge 2$) space.