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.