Perturbation theory of the Stark effect in hydrogen to arbitrarily high order

Abstract

The solution of the Stark effect in hydrogen to arbitrarily high orders of perturbation theory is made feasible by the explicit formula for the $N\mathrm{th}$-order energy in terms of the separation constants through $N\mathrm{th}$ order, derived here. The $N\mathrm{th}$-order separation constant ${\beta }_i^{\mathrm{}\left(N\right)\mathrm{}}$ is shown to be a polynomial of total degree $N+1\mathrm{}$ in the parabolic quantum number $n_i$ and the magnetic quantum number $m$. The polynomial coefficients have been tabulated through seventeenth order and are listed here through tenth order. Similarly, the $N\mathrm{th}$-order energy is a polynomial in the quantum numbers $n_1$, $n_2$, and $m$. The polynomial coefficients (which are more numerous than for ${\beta }_i^{\mathrm{}\left(N\right)\mathrm{}}$) have been tabulated through seventeenth order and are listed here through seventh order. Seventeenth order is high enough to permit a clear numerical demonstration of the asymptotic character of the perturbation series, and a "maximum useful field strength" is defined and illustrated. Energies calculated by perturbation theory for specific states are shown to be in excellent agreement with energies calculated nonperturbatively.