The linear operator
T(f) = f_{xx}f_{yy}f_{zz}2/r f

on smooth functions in R^{3} has eigenfunctions
f_{n,l,m} to the eigenvalues n^{2}. The eigenfunctions
are of the form
f_{n,l,m} = R_{n,l}(r) Y_{l,m}(t,s)

with so called spherical harmonics Y_{l,m}(t,s)=P_{l}^{m}(cos(t)) exp(i m s).
The functions P_{l}^{m}(z) are called Legendre spherical functions
and R_{n,l}(r) are called Laguerre polynomials.
T is the energy operator of the hydrogen atom. For each n,
there are n^{2} eigenfunctions. The function f describes an electron
with energy n^{2}. The energy differences
1/n_{1}^{2}  1/n_{2}^{2}
can literally be "seen". The number n is called the
principal quantum number, the number l is related to the total angular momentum,
and m is related to the zcomponent of the angular momentum.

Lyman  Balmer  Paschen  Bracket

n_{1}=1  n_{1}=2  n_{1}=3  n_{1}=4 
Ultraviolet  visible  infrared  infrared 

