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
for hydrogenic orbitals, namely,
is a Whittaker function that can alternatively be written as
. We can define a radial distribution function (RDF) for a completely filled
. 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.