Wigner–Kirkwood expansion for semi-infinite quantum fluids

Abstract

For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smoothinteractions, the Wigner–Kirkwood formalism provides a semiclassical expansion of theBoltzmann density in configuration space in even powers of the thermal de Broglie wavelengthλ. This result permits one to generate an analogousλ-expansion for the bulk free energy and many-body densities. The present paper brings atechnically non-trivial generalization of the Wigner–Kirkwood technique to semi-infinitequantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to thebulk case, the resulting Boltzmann density also involves position-dependent terms of typeexp(−2x2/λ2) (x denotesthe distance from the wall boundary) which are non-analytic functions of the de Broglie wavelengthλ. Under some condition,the analyticity in λ is restored by integrating the Boltzmann density over configuration space;however, in contrast to the bulk free energy, the semiclassical expansion ofthe surface part of the free energy (surface tension) contains odd powers ofλ, too. Explicit expressions for the leading quantum corrections in the presence of theboundary are given for the one-body and two-body densities. As model systems for explicitcalculations, we use Coulomb fluids, in particular the one-component plasma defined in theν-dimensional(integer ν ≥ 2) space.