Sum Rules for Hydrogenic Wave Functions, with Applications to Charge-Exchange and Ionization Processes

We prove some general addition theorems for certain matrix elements which involve hydrogen-atom wave functions. In particular, if $f_{\mathrm{nlm}}\left(\mathrm{q}\right)$ is the Fourier transform of the hydrogen-atom wave function with quantum numbers $n$, $l$, $m$, then ${\left|f_{\mathrm{nlm}}\right(\mathrm{q}\left)\right|}^2$ summed over all $l$ and $m$ for a given $n$ is equal to $2{}_{}{}^6{}_{}{}^{}\pi {a_0}^{-5\mathrm{}}{n}^{-3\mathrm{}}\times {(q^2+{a_0}^{-2\mathrm{}}{n}^{-2\mathrm{}})}^{-4\mathrm{}}$, where $a_0$ is the Bohr radius. Two applications of the theorems are given. Firstly, we consider charge-exchange reactions of the type ${\mathrm{H}}^{+}+\mathrm{H}\left(n_1{l}_1{m}_1\right)\to \mathrm{H}\left(n_2{l}_2{m}_2\right)+{\mathrm{H}}^{+}$ and use our general theorems to obtain for the cross section for reactions proceeding from the initial atomic state $n_1$ to the final state $n_2$ an expression which is both simple and fully exact (in Born approximation). Secondly, we indicate how the theorems may be applied to get simple expressions for the cross sections for ionization of excited hydrogen atoms by various processes.