Logarithmic perturbation expansions

Abstract

A method previously developed for one-dimensional nonrelativistic perturbation theory is extended to three-dimensional problems. This method essentially consists of performing the perturbation expansion on the logarithm of the wave function instead of on the wave function itself. It is shown that, for the first-order corrections in problems that are not reducible to one dimension, this method is equivalent to that of Sternheimer and to that of Dalgarno and Lewis. In the present approach, the higher-order corrections can be obtained in a hierarchical scheme and there exists an isomorphisim between the equation for the first-order correction and the equation for the $i\mathrm{th}$-order correction. As an illustration of the technique developed, the authors consider the hydrogen atom in an external multipole field and in two different spherically symmetric perturbation potentials, $\ensuremath{\delta}(r\ensuremath{-}a)$ and ${e}^{\ensuremath{-}\ensuremath{\alpha}r}$. The last potential is related to the problem of the screened Coulomb potential. By considering the $\ensuremath{\delta}(r\ensuremath{-}a)$-type potential, two interesting sum rules are obtained.