Semiclassical evaluation of sums of squares of hydrogenic bound-state wavefunctions

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Abstract
The sum over l and m of mod phi nlm(r) mod 2, where phi nlm(r) is a bound-state hydrogenic wavefunction with principal quantum number n(>>1) and angular momentum quantum numbers l and m, is evaluated approximately using the Thomas-Fermi statistical approach. This approach breaks down in the region near the nucleus and in the region near and beyond the classical edge of the nth shell at r=2n2a0/Z, with Z the atomic number. A simple correction, proposed recently by Englert and Schwinger (1984) in the context of heavy atoms, is investigated in the region near the edge. Results are presented. The sum over n and m of mod phi nlm(r) mod 2 and the sum over l and m of mod phi nlm(k) mod 2, where phi nlm(k) is the Fourier transform of phi nlm(r), are also evaluated using the Thomas-Fermi method.