Nonrelativistic Wentzel-Kramers-Brillouin eigenvalues of the Thomas-Fermi neutral-atom potential in the large-atomic-number limit

The Wentzel-Kramers-Brillouin eigenvalue condition is developed in an expansion in $Z^{\mathrm{-}1/3}$ to lowest order in the limit in which the atomic number Z becomes very large. The energy levels are studied explicitly for finite orbital angular momentum l quantum numbers. The nature of the resulting level spectrum is illustrated and its connection with the solutions of Schrödinger’s equation by Latter, for a closely related potential, is briefly discussed. It is pointed out that to get the complete level spectrum near the continuum, for large Z, the case of l of order $Z^{1/3}$ will eventually require consideration. Finally, a few general results are established, one of which predicts the maximum value of l for which a bound state can occur for a given value of Z.