My question is related to the Wikipedia page of spin-forbidden reactions https://en.wikipedia.org/wiki/Spin-forbidden_reactions
When a reaction converts a metal from a singlet to triplet state (or vice versa):
- The energy of the two spin states must be nearly equal, as dictated by temperature,
- A mechanism is required to change spin states.
Strong spin-orbital coupling can satisfy the 2nd condition.
It can be shown that the five operators $H_0$, $J^2$, $L^2$, $S^2$, and $J_z$ all commute with each other and with $\Delta H$.
The above quotation would imply under the spin-orbit coupling operator in the spin-orbit interaction Wikipedia page, $S^2$ is a good quantum number, and the expectation value of $S^2$ will not change, though $S_z$ could change.
Though in Sakurai's Advanced Quantum Mechanics, the author mentioned the Dirac equation for the hydrogen atom, the Hamiltonian commutes with $K, J^2$, and $J_z$, no longer commutes with $L^2$, I assume it won't commute with $S^2$. That leads to my question(s):
Does the spin-forbidden reaction Wikipedia page implies another way of introducing spin-orbit interaction, make $[H, S^2]\neq 0$? Or means $S_z$ is not a good quantum number; Or when people talk about this issue, people imply the full Dirac equation or anything possibly equivalent to it somehow (exact 2-component method)? Or anything else?