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 fₙₗₘ(q) is the Fourier transform of the hydrogen-atom wave function with quantum numbers n, l, m, then |fₙₗₘ(q)|² summed over all l and m for a given n is equal to 2⁶πa₀^-5n^-3×(q²+a₀^-2n^-2)^-4, where a₀ is the Bohr radius. Two applications of the theorems are given. Firstly, we consider charge-exchange reactions of the type H⁺+H(n₁l₁m₁)→H(n₂l₂m₂)+H⁺ and use our general theorems to obtain for the cross section for reactions proceeding from the initial atomic state n₁ to the final state n₂ 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.