
In R^{3} the spherical harmonics correspond to the harmonic
poylnomials that are homogeneous of degree l; we have
dim(H_{l}) = 2l+1 = 1,3,5,7,...
The pure states of a hydrogen atom are given by its principal quantum number
N=1,2,3,... its angular momentum l, and its magnetic quantum number m.
The energy in state N is 1/N^{2}.
Together l and m pick out a basis element Y_{lm} from the
space H_{l}.
These satisfy
l ≤ m ≤ l and 0 ≤ l ≤ N1.
The wave function has the form
f(x,y,z) = e^{r} r^{l} L_{Nl} (r) Y_{ lm } (x,y,z),
where the radial function L_{Nl}(r) is a Laguerre polynomial.
Thus the number of states of hydrogen with energy N is given by
dim(H_{0}) + ... + dim(H_{N1}) = n^{2}.
Traditionally states with values l=0,1,2,3,... are denoted by
s, p, d, f, g, h. Note that a given spherical harmonic in H_{l}
appears for all energies N>l.
(Disclaimer: We ignore the fine structure constant.)
