Generalized Unsöld Theorem for Hydrogenic Functions

According to Unsöld's theorem (see the Demonstration "Unsöld's Theorem"), the sum over all states for an -subshell of hydrogen-like orbitals reduces to a spherically symmetrical function: . For a pure Coulomb potential with any nuclear charge , the different states for a given are also degenerate. The author has derived a generalization of Unsöld's theorem, an explicit form for the sum over both and for hydrogenic orbitals, namely,
,
where is a Whittaker function that can alternatively be written as . We can define a radial distribution function (RDF) for a completely filled -shell by . This is normalized according to , reflecting the orbital degeneracy of the energy level . In this Demonstration the function is plotted for selected values of (1 to 10) and (1 to 25).

L. S. Bartell has derived the classical analog of , which, in accordance with Bohr's correspondence principle, approaches the quantum result in the limit . The checkbox produces a red plot of the classical function.

The generalized Unsöld theorem has found several theoretical applications, including derivation of the canonical Coulomb partition function, density-functional computations, supersymmetry, and study of high- Rydberg states of atoms.