Semiclassical densities in phase space: Wigner's distribution function for thermodynamic equilibrium

Abstract

The quantum-mechanical joint (quasi-) density in phase space (the Wigner distribution) for thermodynamic equilibrium is studied in the semiclassical limit h(cross) to 0. The approximation is of an asymptotic (WKB) type and nonperturbative in character. The equilibrium phase-space distribution can be expressed in terms of classical paths satisfying certain well defined conditions. These paths are complex-valued, i.e. classically forbidden. The resulting semiclassical expression for the equilibrium Wigner phase-space density agrees with the exact quantum result for the analytical solvable case of harmonic potentials for all temperatures.