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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.

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  • $\begingroup$ I can't speak for your specific Hamiltonian, but the general answer must be 'no', since good quantum numbers are classically conserved quantities and if it were easy to identify conserved quantities physics would be a lot easier. $\endgroup$
    – jacob1729
    Jul 22, 2019 at 18:54
  • $\begingroup$ All four $S_i\cdot S_i$ commute with the Hamiltonian, here, no? $\endgroup$ Jul 22, 2019 at 19:46
  • $\begingroup$ Cosmos, you are right. and Sorry that was a typo. $\endgroup$
    – user193422
    Jul 22, 2019 at 20:48

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