Theory of the ionization of the hydrogen atom by an external electrostatic field

The theory of the field ionization of the hydrogen atom is developed analytically. The leading term of an asymptotic expansion for the ionization rate—the reciprocal lifetime—is derived. In the weak-field limit, the formula for the ionization rate reduces to $\frac{1}{\tau }=n^{-3\mathrm{}}{[n_2!(n_2+\left|m\right|)!]}^{-1\mathrm{}} {\left(n\frac{{}_{}{}^3{}_{}{}^{}F}{4}\right)}^{-2\mathrm{}{n}_2-{\mathrm{}\left|m\right|\mathrm{}}_{-1\mathrm{}}}\times \exp \left[3\right(n_1-n_2\left)\frac{-2\mathrm{}}{\left(3n{}_{}{}^3{}_{}{}^{}F\right)}\right]$, where $n$, $m$, and $n_1$ and $n_2$ are the usual principal, magnetic, and parabolic quantum numbers, respectively, and $F$ is the field strength in atomic units. For the ground state, this formula agrees with that of Landau and Lifshitz. For all $m^2=1\mathrm{}$ states, the formula agrees asymptotically for large $n_2$ with that of Lanczos after correction of the latter for an error. It is in disagreement with the result of Oppenheimer and with the low-field result of Rice and Good. Significantly better agreement with numerical calculations of Alexander, of Hehenberger, McIntosh, and Brändas, of Damburg and Kolosov, and of Bailey, Hiskes, and Riviere is obtained with a formula for not quite such small $F$, $\frac{1}{\tau }={(-2\mathrm{}\mathrm{Re}E)}^{\frac{3}{2}}{[n_2!(n_2+\left|m\right|)!]}^{-1\mathrm{}}{f}^{-B}\exp [-\frac{1}{\mathrm{}\left(6f\right)\mathrm{}}]$, where $E$ is the perturbed energy, where $f=\frac{\left[{\mathrm{}(-2E)\mathrm{}}^{-\frac{3}{2}}F\right]}{4}$, and where $\frac{B}{2}={\beta }_{2,n_2}$ is the usual perturbed separation constant [${\beta }_2\to \frac{n_2+\left|m\right|\mathrm{}}{2}+\frac{1}{2}$, as $F\to 0$].