Generalized Unsöld Theorem for Hydrogenic Functions
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: (r,θ,ϕ)=(r). 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,(r)=(r)=(2Zr/n)-(2Zr/n)(2Zr/n),where (z) is a Whittaker function that can alternatively be written as . We can define a radial distribution function (RDF) for a completely filled -shell by (r)=4π(r). This is normalized according to (r)dr=, reflecting the orbital degeneracy of the energy level . In this Demonstration the function (r) is plotted for selected values of (1 to 10) and (1 to 25).
m
l
l
∑
m=-l
ψ
nlm
2
|
ρ
nl
Z
l
n
l
m
n-1
∑
l=0
ρ
nl
ρ
n
3
Z
π
3
n
2
M
n,1/2
M
n,1/2
''
M
n,1/2
M
m,1/2
z(1-n;2;z)
-z/2
e
1
F
1
n
D
n
2
r
ρ
n
∞
∫
0
D
n
2
n
E
n
D
n
Z
n
L. S. Bartell has derived the classical analog of (r), 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.
D
n
n∞
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.
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