On the sign of the second-order energy shift in the Rayleigh–Schrödinger perturbation theory for a highly excited state

Abstract

The second-order energy shift in the Rayleigh–Schrödinger perturbation theory is most likely to be positive for large quantum numbers, i.e. for excited states. Within the Wilson–Sommerfeld approximation, good for large quantum numbers, we show on the contrary that the sufficient condition for the second-order shift to be negative is, that the total potential be symmetric and both the unperturbed and perturbed potentials be monotonically increasing.