On the classical approximation involved in the Thomas-Fermi theory

Abstract

The nature of the expansion of the particle density of a system in powers of h is investigated. An exact expression for the density in a system with an harmonic potential is obtained. There are two types of terms. First, there are steady terms which agree with those derived by other authors and yield in first approximation the Thomas-Fermi result. Secondly, there are oscillatory terms analogous to the de Haas-van Alphen terms in the susceptibility of an electron gas and which cannot be expanded in powers of h . For systems with an odd number of dimensions the series of steady terms is shown to be divergent for all values of h =(= 0. If the number of dimensions is even there is only a finite number of steady terms.