High-impact-velocity forward charge transfer from high-Rydberg states as a classical process

It was long ago suggested by Thomas that the charge-transfer cross section in the neighborhood of forward scattering was dominated by a double-scattering process. Thomas's analysis, which was almost completely classical, might well suggest that, in a proper quantum-mechanical study, the second Born contribution would dominate over the first, and this was ultimately found to be the case. Unfortunately, the problem which Thomas considered was charge transfer to any bound state of an incident proton from a hydrogen atom in its ground state, a problem which cannot truly be studied classically. Thomas found that the cross section behaved as $r^{-\frac{7}{2}}$, where $r$ is the initial electron-target-proton separation, and simply replaced $r^{-\frac{7}{2}}$ by $a_0^{-\frac{7}{2}}$, where $a_0$ is the Bohr radius. The result he obtained is identical in form to that obtained in the second Born approximation, but the coefficient is smaller by about a factor of ten. The more consistent procedure of replacing $r^{-\frac{7}{2}}$ by its expectation value gives $\infty$ for the $1s$ ground state, (or indeed for any state with orbital angular momentum quantum number $l=0\mathrm{}$). We show that for capture from a high-Rydberg state, that is, a state with principal quantum number $n\gg 1$, the classical picture is not only meaningful for $l\ne 0$, but, for $l$ sufficiently large, becomes exact.