# Hyperfine structure

In Griffiths, the hyperfine structure is described as follows:

So the hyperfine structure is a result of a mechanism called spin-spin coupling, which is the interaction of the spin of the nucleus (proton in this case) with the spin of the electron. The spin of the nucleus has a magnetic moment which creates a magnetic field and this magnetic field creates a torque on the spin/magnetic moment of the electron. So this lifts the degeneracy in the triplet state and the singlet state.

However, when I look up another image, I get the following:

where $$F = J + S$$, where $$J$$ is the total angular momentum of the electron (spin and orbital angular momentum) and $$S$$ is the spin of the nucleus/proton. This lifts the degeneracy in the $$m_j$$ levels (if that is even a correct statement).

So my questions are:

• Are these two things consistent with each other? (degeneracy in spin states vs. degeneracy in mj levels).

• Is it even correct to state that hyperfine structure lifts the degeneracy in mj levels? (This is however true for the Zeeman effect), as there is a clear dependence on mj)

• Is the hyperfine structure a result of the interaction between the spin of the nucleus and the spin of the electron (spin-spin coupling)? Or is it a result of the interaction between the spin of the nucleus and the total angular momentum of the electron (so spin of electron and orbital angular momentum of electron)?

• Note that the calculation done in Griffiths is only for the ground state $l=0$ case in hydrogen. Therefore, orbital effects are irrelevant. The more general statement is that the total nuclear spin interacts with the total angular momentum of the electron. See the wikipedia page on hyperfine structure for some more detail. (And, if I remember correctly, Griffiths starts with a somewhat more general case but argues that for the $l=0$ state, most of the other stuff averages out.) Commented May 20, 2023 at 17:43
• Oh I see, so spin-spin coupling is only relevant for the hyperfine splitting for the ground state and not generally speaking? Commented May 20, 2023 at 17:54

Griffiths only calculates the hyperfine splitting for the ground state. However, as he states, his result can be easily generalized for any state with l=0. So, in the ground state, $$J = \frac{1}{2}$$ since l=0.
• In general, it should since there is a term ~$$\vec L$$. But keep in mind that in many cases already the fine structure does that.
• As mentioned above, in case of $$l \neq 0$$ there is a term in the hyperfine splitting that contains $$\vec L$$. So it is not only a result of the interaction between the spin of the nucleus and the spin of the electron.
For the two $$^2P_J$$ states, the proton spin may split the $$J=1/2$$ state into a singlet or a triplet of hyperfine states. But the proton can’t “cancel” the $$J=3/2$$ state into a singlet; its options are the $$F=1$$ triplet or the $$F=2$$ quintuplet.