Semiclassical expansion theory in phase space.

Abstract

We study the \ensuremath{\Elzxh} perturbation expansion of the quantum Wigner equation. It leads to a unified formulation of semiclassical approximations based on the phase-space representation of quantum mechanics. We derive the O(${\mathrm{\ensuremath{\Elzxh}}}^{2}$) quantum corrections to the finite-temperature Bose and Thomas-Fermi phase-space distributions. Both reduce, in the high-temperature limit, to the known quantum corrections of the classical Gibbs-Boltzmann probability density. Within this approach, moreover, we obtain a very simple derivation of the extended Thomas-Fermi theory. Finally, the limits of applicability, the convergence problems, and the possibility of improving or defining new semiclassical approximations are discussed.