Generalizations of the Fermi-Segrè formula.

Abstract

Phase-integral formulas obtained by Fr\oman, Fr\oman, and the present author for the limit of ${u}_{N}$(l,E,r)/${r}^{l+1}$ as r\ensuremath{\rightarrow}0, where ${u}_{N}$(l,E,r) is a normalized nonrelativistic wave function for bound or unbound states, are compared with corresponding formulas obtained by Pratt and co-workers from analytical perturbation theory. It is demonstrated that the phase-integral formulas are practically always advantageous to the perturbation-theory formulas. If one substitutes the perturbation-theory approximation for the energy into the respective phase-integral formula for the limit of ${u}_{N}$/${r}^{l+1}$ as r\ensuremath{\rightarrow}0 and performs approximations according to a well-defined scheme, the approximate perturbation-theory formulas for the limit of ${u}_{N}$/${r}^{l+1}$ as r\ensuremath{\rightarrow}0 are obtained. For bound s states the phase-integral formula and the perturbation-theory formula coincide with the Fermi-Segr\`e formula in the nonrelativistic approximation.