Central Fields and Rydberg Formulas in Wave Mechanics

The analogy in wave mechanics is set up for the non-penetrating part of the electron orbits on the older theory, in which the motion of the outer electron is like that in the field of a point charge. The Rydberg formula follows from this as in the older theory. The separate Rydberg series for the various multiplets also are explained. The method previously applied by the writer to the helium problem appears as a next approximation, in which the outer electron moves, not in a hydrogen-like field, but in a somewhat more accurate central field. It is shown that this method should give a good approximation to the energy levels, and also to the wave function, except at very small distances. The attempt to use this function as a starting point for perturbation calculations is shown, however, not to be justified, for the matrix of the Hamiltonian function $H$, which is used in the perturbation method, greatly magnifies the errors in the wave function. Suggestions are made as to the next step in improving the method, by increasing the accuracy of the wave function for small $r\text{'}\mathrm{s}$.