Partial resummation of the Wigner—Kirkwood expansion for quantum Boltzmann densities

Abstract

The logarithmic Wigner—Kirkwood (WK) expansion for the one-dimensional quantum Boltzmann density < X|exp[−β( p 2/2 m + V)]| X> is resummed over powers of V′'( X) = d 2 V/d X 2. The resummed expansion is as simple to use as the original WK expansion, but stays usable down to zero temperature where V′'( X) > 0. In lowest order, it yields an approximation initially introduced by Miller.