The **interaction between the spin of the nucleus and the angular momentum of the electron**
causes a further
(hyperfine) splitting
of atomic states.
It is called hyperfine because it is also order
like the fine structure corrections, but it is smaller
by a factor of about
because of the mass dependence of the spin magnetic moment for particles.

The magnetic moment of the nucleus is

where is the

We computed the hyperfine contribution to the Hamiltonian for
states.

Now, just as in the case of the , spin-orbit interaction, we will define the total angular momentum

It is in the states of definite and that the hyperfine perturbation will be diagonal. In essence, we are doing degenerate state perturbation theory. We could diagonalize the 4 by 4 matrix for the perturbation to solve the problem or we can use what we know to pick the right states to start with. Again like the spin orbit interaction, the

For the hydrogen ground state we are just adding two spin particles so the possible values are . The transition between the two states gives rise to EM waves with cm.

We will work out the
**effect of an external B field on the Hydrogen hyperfine states**both in the strong field and in the weak field approximation.
We also work the problem without a field strength approximation.
The **always applicable intermediate field strength result**
is that the four states have energies which depend on the strength of the B field.
Two of the energy eigenstates mix in a way that also depends on B.
The four energies are

Jim Branson 2013-04-22