Good quantum number for the weak field Zeeman effect

In the presence of a magnetic field $vec B$, the Hamiltonian of this system is given by

$$H = -(vec mu_l + vec mu_s). vec B$$

where $vec mu_s$ and $vec mu_l$ are the magnetic dipole moments for spin and orbital angular momentum respectively.

A “good set” of unperturbed energy eigenstates is given by $| n l j m_j rangle$ and $m_l$ and $m_s$ are considered “not good” quantum numbers, since in the presence of spin-orbit coupling, the Hamiltonian $H$ does not commute with the operators $L$ and $S$. Operators that commute have computable simultaneous eigenvalues.

Note that in the strong-field limit, the unperturbed Hamiltonian does commute with $L^2$, $S^2$, $L_z$ and $S_z$, and so a “good set” of unperturbed energy eigenstates is given by $|n l m_l m_s rangle$.

Probably funny terminology, but that is probably what the author is getting at.