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Dos the total angular momentum, $J=S+L$, commute with the hamiltonian of a general sistem, with no particularities?

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No, it doesn't. There are plenty of systems whose hamiltonian does not commute with the total angular momentum.

For an isolated system the result does hold, but if there's any anisotropic interaction with the outside world (say, an external potential) then it will break.

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