One way to break the spherical symmetry is to apply an external B field.
Lets assume that the field is weak enough that the energy shifts due to it are
smaller than the fine structure corrections.
Our Hamiltonian can now be written as
, where
is the normal Hydrogen problem,
is the fine structure correction, and
We now run into a problem because picks eigenstates of and while picks eigenstates of and . In the weak field limit, we can do perturbation theory using the states of definite . A direct calculation of the Anomalous Zeeman Effect gives the energy shifts in a weak B field.
The factor is known as the Lande Factorbecause the state splits as if it had this gyromagnetic ratio. We know that it is in fact a combination of the orbital and spin g factors in a state of definite .
In the strong field limit we could use states of definite
and
and
calculate the effects of the fine structure,
, as a correction.
We will not calculate this here.
If the field is very strong, we can neglect the fine structure entirely.
Then the calculation is easy.
Jim Branson 2013-04-22