Cumulant approximations and renormalized Wigner-Kirkwood expansion for quantum Boltzmann densities

The logarithm of the quantum Boltzmann density &‖X〉, where $H_V$=$p^2$/2m+V, is expressed as a cumulant expansion in powers of v=V-W, where W(x)=V(X)+V’(X)(x-X)+(1/2)V’’(X )(x-X${)}^2$ is the local quadratic approximant to V(x) at the point X. Where V’’(X)>0, this expansion behaves ‘‘nonsecularly’’ as β→∞ (all its terms ∼β), and thus remains a useful approximation scheme even as the temperature ${\beta }^{\mathrm{-}1}$→0 (in that limit, it yields Rayleigh-Schrödinger perturbation expansions of the ground state of $H_V$). By Taylor expanding v(x) about X in the cumulant expansion, we obtain an expansion which is a resummation over powers of V’’(X) of the Wigner-Kirkwood (WK) expansion of ln${\rho }_V$; this ‘‘renormalized’’ WK expansion, whose coefficients are simple functions of V’’(X), is as simple to use as the ordinary WK expansion, yet more accurate where V’’(X)≠0, and usable down to zero temperature where V’’(X)>0 (yielding, in that limit, WK-type expansions for the ground state of $H_V$). In lowest order, it yields an approximation initially proposed by Miller.