Zeeman effect, something I'm not understanding, about history

[fluidistic]

With the normal Zeeman effect, I think the splitting of the emission/absorbtion lines is worth $\Delta E = \mu _B B m_l$. That was before they knew about the spin.
When they discovered the spin they realized that in fact the energy splitting was worth $\Delta E =g_l \mu _B B m_l$ where $g_l=1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}$.
Now for example if I take the hydrogen atom in its ground state, $g_l=2$. So that the $\Delta E$ is twice as big as what they thought it was before the understanding of the spin.
How could they think that their formula before the spin introduction was "ok"? I mean a factor 2 looks enormous to me. Am I missing something?
Besides, why should one use the formula for the normal Zeeman effect in -undergraduate physics- problems while it doesn't seem (at least to me) give any value close to the real ones? I feel like I'm really missing something.
Can someone shed some light on this?

 

[jtbell]

Not knowing anything about spin at first, people tried to predict the Zeeman effect by using only the orbital angular momentum, and came up with your first equation. Sometimes this prediction actually agrees with experiment, and this was called the "normal Zeeman effect." Sometimes (usually, in fact!) it doesn't agree with experiment. This was called the "anomalous Zeeman effect," and people looked for ways to explain it.

Then spin was discovered, people re-did the Zeeman effect prediction taking spin into account, and got your second formula. This prediction does agree with the observed "anomalous Zeeman effect."

It turns out, as I recall, that the "normal Zeeman effect" is actually something of an exceptional situation in which the spin effects fortuitously cancel out in some multi-electron atoms. So the "normal Zeeman effect" is really anomalous, and the "anomalous Zeeman effect" is really normal.

 

[fluidistic]

Ah ok... also I think I made a mistake in the second formula. It's $m_j$ instead of $m_l$. And for the hydrogen atom in ground state, $m_j=m_s$ which can be either -1/2 or 1/2.
So what I originally thought was a factor 2 is in fact a factor 1. In other words the normal and anomalous Zeeman effect are the same... well I think so, if I applied the good formulae. So I am in one of these situations you describe.
Thanks.