What do the dashed lines represent in this figure from the discussion of the Zeeman effect in Griffiths? Consider the figure below (figure 6.12 from Griffiths, Introduction to Quantum Mechanics, p 249 in the 1995 edition), which shows the Zeeman effect on the $n=2$ eigenvalues of hydrogen. The figure shows two dashed lines. What's the significance of dashed lines? I mean, why are they dashed while others are not?



 A: There isn't enough mention of those lines in the text to be able to tell for sure, but the obvious interpretation is that the dashed lines are only there as a guide to the eye so that it is easier to see that the second and sixth eigenvalues (either order) in the strong-field regime scale linearly with $B_\mathrm{ext}$ with a nonzero offset. This would be clearer if the figure included the explicit offset into the asymptote, as in the figure below, but we can't always account for other people's choices.


Image source

A: I tend to agree with what @Emilio Pisanty said, but there is some subtlety involving the n=2 level of hydrogen. Namely, according to relativistic quantum mechanics, ${}^{2}S_{1/2}$ and ${}^{2} P_{1/2}$ of hydrogen are degenerate up to fine structure. The degeneracy is broken by the Lamb shift, which needs QED to explain. More info on that here. The degeneracy is really special to hydrogen because of a hidden symmetry (search "Runge-Lenz vector").  But this Lamb shift is very tiny so you won't be able to see it clearly on the y-axis of the diagram at $B=0$. 
So what I want to say is that the two dashed lines correspond to magnetic sublevels of ${}^{2} S_{1/2}$, and for the thick lines, the upper four lines correspond to ${}^{2} P_{3/2}$ and the lower two to ${}^{2} P_{1/2}$. 
I will give a long explanation of what is going on in the diagram. Maybe it's already explained in the textbook and you already know, but I'll do it for consistency. 
In both low- and the high-field region, the energy shifts of the magnetic sublevels are (approximately) linear with respect to magnetic field. This is because in those limits you have well-defined quantum numbers (conserved quantities)


*

*In the low-field region, the orbital angular momentum $\vec{L}$ and the spin angular momentum $\vec{S}$ combine to form the the total angular momentum $\vec{J} = \vec{L}+\vec{S}$, and $\vec{J}$ precesses around the internal, effective magnetic field created by spin-orbit coupling. The Zeeman shift is $\Delta E \approx -g_J \mu_B \vec{J}\cdot \vec{B}_{\text{ext}}$, where $g_J$ is the g-factor of the total angular momentum, $\mu_B$ is the Bohr magneton, and $B_{\text{ext}}$ is the external magnetic field. To find $g_J$, look up Landé g-factor.

*In the high-field region, when the external magnetic field is much stronger than the internal, effective magnetic field, $\vec{L}$ and $\vec{S}$ precess around the external magnetic field separately and they are decoupled. The Zeeman shift is $\Delta E \approx - (g_L \vec{L} + g_S \vec{S})\cdot \mu_B\vec{B}_{\text{ext}}$

*To find the exact solution that describes both limits and the intermediate region, use Breit-Rabi formula. Wikipedia article on Zeeman effect has a derivation of this. 


Now let's describe what is happening in the diagram you shared. In the low-field region, you see two branches, with the upper branch having four lines and the lower branch having two. The upper branch corresponds to ${}^{2S + 1}L_{J} = {}^2 P_{3/2}$ and the lower branch corresponds to ${}^{2} P_{1/2}$. But there is also the degenerate ${}^{2} S_{1/2}$ level. You see four sublevels emerging from the upper branch, and that makes sense because $J = 3/2$ has $2J +1 = 4$ sublevels. The lower branch should have 2 sublevels accordingly. 
But shouldn't we be able to distinguish ${}^{2} S_{1/2}$ and ${}^{2} P_{1/2}$ when the degeneracy is broken by the external magnetic field? Indeed, if you calculate the Lande $g_J$, for ${}^{2} S_{1/2}$ you get $g_J = g_S = 2$ and for  ${}^{2} P_{1/2}$ you get $g_J = 2/3$. So what I am suggesting is that those dashed lines correspond to ${}^{2} S_{1/2}$ and the thick lines correspond to ${}^{2} P_{1/2,3/2}$.
Now let's check if the slopes of the lines make sense. Let's focus on the bottom two lines, at the high-field limit. The dashed line has roughly half the slope of the bottom-most line. In the high-field limit, the bottom-most line corresponds to $|L = 1, m_L = -1 \rangle |S = 1/2, m_S = 1/2 \rangle$. To get the right sign for $m_L$ and $m_S$, consult the signs of the g-factors. The energy shift slope should be $1 \cdot 1 + 2 \cdot 1/2 = 2$. On the other hand, the dashed line should have slope of $2 \cdot 1/2 = 1$. 
A: The dashed lines are the ones that arise from the S state (n=2, l=0, j=1/2).
The other lines all arise from P states. 
This graph ignores the lamb shift. 
According to Dirac (not quantizing the EM field), the state (n=2, l=1, j=1/2) has the same zero-field energy, splits also in two at non-zero fields, but has a smaller g-factor and thus smaller splitting. 
The remaining lines at the top arise from the state (n=2, l=1, j=3/2), which splits into 4 lines.
In the low field limit, the g-factors are g_j = [3j(j+1)-l(l+1)+s(s+1)] / [2j(j+1)].
If I am not mistaken, these are 
g_3/2 = 4/3
g_1/2 = 2/3 for the P state
g_1/2 = 2 for the (dashed) S state
In the high field limit, it can be clearly seen, how spin and orbital angular momentum decouple. The top half is spin up, the lower half spin down. In each half one can see 4 l states, namely (l=0, m_l=0) and (l=1, m_l=-1,0,+1).
