Hydrogenic Stark effect: Properties of the wave functions

Abstract

A modified WKB treatment of a hydrogenic atom in a static electric field $F$ is found accurate over all energies $\ensuremath{\epsilon}$ to within 1% for $F\ensuremath{\le}5000$ kV/cm. This semianalytical technique describes the two properties of the wave function most relevant to photoionization and scattering: (1) the ratio of amplitudes at large and small distances ${A}_{{n}_{1}m}(\ensuremath{\epsilon},F)$, and (2) the asymptotic phase shift ${\ensuremath{\delta}}_{{n}_{1}m}(\ensuremath{\epsilon},F)$. The photoionization cross section from the ground state, given as a semianalytical function of $\ensuremath{\epsilon}$ and of the quantum numbers $m$ and ${n}_{1}$, describes the polarization-dependent shape resonances observed above the zero-field ionization threshold ($\ensuremath{\epsilon}=0$) and agrees with the exact calculations of Luc-Koenig and Bachelier. Complex-contour integration of WKB phase integrals reveals an equipartition of phase between the two sides of the potential barrier formed by the Coulomb and Stark fields. The total phase shift increases by $2\ensuremath{\pi}$ (not $\ensuremath{\pi}$) between successive quasistationary levels of constant ${n}_{1},m,F$. The WKB method originally applied to the Stark effect by Lanczos and generalized by Miller and Good is reviewed in an appendix.