The relation between the Wentzel-Kramers-Brillouin and the Thomas-Fermi approximations

Abstract

It is shown that two essential approximations are made in using the customary Thomas-Fermi formula for the sum of the eigenvalues in any one-dimensional problem. The first is to start from the Wentzel-Kramers-Brillouin formula for the individual eigenvalues, and the second is to replace the summation by an integration. The three-dimensional central field problem is then considered, and by similar arguments, though with an additional approximation, the usual Thomas-Fermi energy formula is again obtained. Possible ways of correcting the errors introduced by integrating instead of summing are discussed and illustrative examples given. In the three-dimensional case particular attention is given to the Coulomb field problem. Finally, brief reference is made to the possibility of correcting for the errors of the Wentzel-Kramers-Brillouin formula.