Linear response of the hydrogen atom in Stark states to a harmonic uniform electric field.

Abstract

The influence of a weak harmonic uniform electric field, switched on adiabatically, on a nonrelativistic hydrogenlike atom is examined. Each of the \ensuremath{\varphi}- and A-gauge first-order corrections to the wave function of a stationary state \ensuremath{\Vert}N〉 is determined by a vector function that we denote ${\mathrm{v}}_{N}$ and ${\mathrm{w}}_{N}$, respectively. The absolute starting point of our calculations is Schwinger's formula for the Coulomb Green's function in momentum space. In the case of a bound state with definite angular momentum, we report a compact integral representation and also an explicit expression of the \ensuremath{\varphi}-gauge vector ${\mathrm{v}}_{\mathrm{nlm}}$, which are analogous to those of the corresponding A-gauge vector ${\mathrm{w}}_{\mathrm{nlm}}$ studied previously. We have derived compact analytic expressions of the linear-response vectors ${\mathrm{v}}_{{n}_{\ensuremath{\xi}}{n}_{\ensuremath{\eta}}m}$ and ${\mathrm{w}}_{{n}_{\ensuremath{\xi}}{n}_{\ensuremath{\eta}}m}$ associated to an arbitrary Stark state. These are written first as contour integrals, and then explicitly in terms of a new generalized hypergeometric function with five variables, $_{2}\mathrm{\ensuremath{\varphi}}_{\mathrm{H}}$, which is a finite sum of Humbert functions ${\ensuremath{\varphi}}_{1}$. We have calculated the static limit of the regular part of the vector ${\mathrm{v}}_{{n}_{\ensuremath{\xi}}{n}_{\ensuremath{\eta}}m}$. Also discussed are the Sturmian-function expansions of the linear-response vectors for angular momentum states.