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
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
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