Formalism of distribution-function method in impurity screening

Using the Weyl-Wigner formulation of quantum mechanics, a rigorous quantum-mechanical basis of the distribution-function method in potential screening, both with and without an externally applied magnetic field, is given. An expansion of the local particle density, up to order ${\hslash }^2$, Leads to a quantum correction of the Thomas-Fermi self-consistent potential approximation and to the so-called "quasiclassical approximation" for nonzero magnetic field. The important results are nonlinear in the potential characteristic of the distribution-function method. They are compared with the results of Horing, based on linear response and random-phase approximation using Green's function, for the low- and very-high-field regime, thus, clarifying the exact nature of the "quasiclassical approximation" as a leading term (zero order in h) in a very-high-magnetic-field approximation. The formalism sheds some light on some of the unfamiliar aspects of the results based on linear response.