Uniform semiclassical approximation to the electron density distribution

The single particle reduced density matrices for fermions in a linear potential in one and three dimensions are derived. An idempotent reduced density matrix for a nonuniform field with a single turning point is obtained. The density expression contains the contributions from the direct path and the path with one turning point. When applied to a potential well the corrections from the long-time paths are taken from the theory of Light and Yuan. Only classical quantities of momentum, time, and action enter in the final expression. The density distribution for the nonclassical region is more accurate than that given by the formula of Light and Yuan