Labelling hyperfine structure states in strong magnetic field I am trying to work out the frequency shifts to the hyperfine energy levels in $^{39}$K $\,$ S$_{1/2}$ (the ground state).
I diagonalise the Hamiltonian for different values of the $B_z$ field, with a basis that is an eigenstate of $$ \hat{\mathbf{L}}\,\otimes \hat{\mathbf{S}}\,\otimes\hat{\mathbf{I}},$$ these being the orbital, spin and nuclear angular momenta respectively.
I get something like this:
Which looks qualitatevely correct.
Question: how would I label these states? 
The eigenvectors corresposing to the eigenvalues in the graph are $B_z$ dependent, so I guess I can't really use $m_F$ as a good quantum number. But that's what is usually done in textbooks, along with a $m_J$ number as well. How would I get these from my eigenstates?
Also, 2 of the e-states are $B_z$ independent (the orange and the dark blue one)... is there a physical interpretation for this?
 A: First of all, I think you have a sign error and your manifolds should actually be flipped around. Oops! This is easy to do if you use the wrong sign for $g_I$ and $g_J,$ and different authors use different conventions about where these signs go.
The convention is to label the eigenstates by the state they are adiabatically connected to at zero field. This means labelling them by $(F,m_F)$, where $F$ and $m_F$ are the total angular momentum and the projection of the total angular momentum along the magnetic field.
The $F$ label corresponds to the hyperfine manifold, which is $F=2$ for the five states that go to the same energy at zero field, and $F=1$ for the other three. Then, for each manifold, you can find the $m_F$ number by looking at the low field limit where they are all linear. In this regime,
$$U=g_F m_F B$$
, so for a given $F$ and $B$ the $m_F$ states go in order from $m_F=-F$ to $m_F=F$.
$g_F$ is the total angular momentum gyromagnetic ratio, which can be calculated from $g_I$ and $g_J$ as:
$g_F\sim g_J \frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)} $$
again being careful about the sign convention used  (1).

Finally, if you feel like cheating, you can find the correct diagram here.
