Few-electron atoms with linear Bohr–Sommerfeld electron paths

Abstract

With a pedagogical aim suited for the upper-division undergraduate, we apply the old quantum theory (pre-Schrödinger) to the study of many-electron atomic species. We eschew the typical picture with circular atomic Bohr orbits of non-zero angular momentum and instead consider the electrons to be ‘bouncing’ along straight lines on the nucleus. Abandoning the circular orbits of Bohr comes at the cost of a meanfield approximation but at the gain of a physically correct (vanishing) electron angular momentum for the first four elements. The Bohr–Sommerfeld meanfield (or perturbation) calculations, of which we present a variety of increasing numerical complexity, generally give results accurate to within a few percent. For He, also excited states are calculated and these results quickly converge on the exact values already for the first excited state. The main source of error in the semiclassical calculation with respect to the exact results is traced to the neglect of the Pauli principle, since it is virtually present only for the singlet ground-state but not the lowest triplet state.