The primary reason asking this question to understand good quantum number from a giver Hamiltonian. Is there any good approach that we can identify them?
For example: We have a square and in that four corners there are four spin $J$ are located. Where $J>1$ and they spins interact antiferromagnetically. The hamiltonian is $J(S_1 \cdot S_2 + S_2 \cdot S_3 + S_3 \cdot S_4 ) +S_4 \cdot S_1$.
UPDATE:
My guess is to check the commutation relation hamiltonian and any conserve quantity i want to check. But it seems like a validating a conserved quantity if that is a good quantum number or not.