What are the 'good' quantum numbers for the weak- and strong-field Zeeman effect? I'm quite confused on the 'good' quantum numbers. I thought the good quantum numbers could be defined as the quantum numbers which corresponding operators commute with each other and the Hamiltonian. This definition seemed to work for me but the Zeeman effect is an exception.
First of all the Hamiltonian for the Zeeman effect is  $H'_z = (L + 2S)eB/2m$
In the case of the weak-field zeeman effect we can treat the zeeman effect as a perturbation and they choose n, L, J and $m_j$ as the good quantum numbers. This seems logical to me because each of the corresponding operator commutes with the Hamiltonian as far as I know.
Now for the strong-field Zeeman effect they take S, L, $m_l$ and $m_s$ as the good quantum numbers. Why is that? The hamiltonian didn't change? $L_z$ doesn't commute with L right?
 A: I refer to the answer that I have posted here for additional context.
To commute with the Hamiltonian, in simpler words, is a constant with time.

First of all the Hamiltonian for the Zeeman effect is $H_z' = \frac{eB_{\text{ext}}}{2m} \left( L + 2S \right)$

Note that this is the dominating pertubing Hamiltonian for the strong-field Zeeman Effect, otherwise known as the Pachen-Back Effect, since the external $B$ field dominates over the internal $B$ field, and so, total angular momentum is not conserved, and hence $j$ and $m_j$ are not good quantum numbers, they are not constant with time. We say that the fine structure is treated as a pertubation over the external magentic field.
For the weak-field Zeeman effect, we should treat the external magnetic field as a pertubation over the fine structure Hamiltonian. Indeed, $j$, $m_j$ will be conserved but not $s$, $m_s$, $l$ and $m_l$.

Edit 1: Take the following Hamiltonians, $$ H_0 = \frac{P^2}{2m} - \frac{e^2}{4\pi\varepsilon_0}\frac{1}{r}, \\ H_{\text{f.s.}} = \underbrace{-\frac{P^4}{8m^3c^2}}_{\text{Relativistic Correction}} + \underbrace{\frac{1}{2m^2c^2}\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r^3}}_{\text{Spin-Orbit Correction}} + \underbrace{\frac{\pi \hbar^2}{2m^2c^2}\frac{e^2}{4\pi\varepsilon_0}\delta^3\left(r\right)}_{\text{Darwin Term}},\\ H_{\text{Zeeman}} = -\left(\vec{\mu}_l + \vec{\mu}_s \right)\cdot \vec{B}_{\text{ext}} = \frac{eB_{\text{ext}}}{2m} \left( L + g_SS \right).  $$ and apply pertubation theory in the given orders depending on the cases.  
