Stark Effect in Hydrogenic Atoms: Comparison of Fourth-Order Perturbation Theory with WKB Approximation

Abstract

The Schrödinger equation for a hydrogenic atom in a uniform electric field is separated in parabolic coordinates, and WKB quantization rules are obtained for two of the separated equations. In the weak-field limit, the quantization rules are expanded in powers of the field and evaluated analytically. From the results, a power series for the energy eigenvalues is obtained which agrees with all known results of perturbation theory to fourth order. For strong fields, the quantization rules are evaluated numerically for states for which the perturbation series is expected to be least accurate. It is found that the perturbation results to fourth order agree with the numerical results for the energy eigenvalues to one part in a thousand for the highest fields possible, and that the addition of the fourth order in perturbation theory improves the accuracy of the perturbation series, contrary to the conclusions of an earlier work. Finally, the accuracy for the measurement of fields from observed Stark shifts by using the perturbation theory is estimated.