If $[S_i,H]\neq0$ or $[S^2,H]\neq0$, we might say spin is not a good quantum number in the system. But is there any more practical or detailed criterion? Or certain families of magnetic interaction forms always make it not conserved?

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    $\begingroup$ You seem to say that whether $S_i$ or $S^2$ commute with H it's somehow the same thing and it's not. These two cases describe different conservation laws. Commutation with H is the most concise and general way to capture conserved quantities ("good quantum numbers") that I know of; anything else is likely going to be applicable only to specialized Hamiltonians and even then I doubt you can do better. $\endgroup$ – oleg Jul 29 at 23:58
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    $\begingroup$ This is not a complete answer, but have you heard a spin-orbit interaction ($ \propto L \cdot S $) in which S is no longer a good quantum number but the total ($J$) is. So the point is that whenever you another form of angular momenta where they can "interact" with spin, the spin is no longer a good quantum number. $\endgroup$ – rnels12 Jul 30 at 6:15

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