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Suppose J1 and J2 are angular momentum of two different particles or different kinds of for same particle then we define coupled and uncoupled basis for them and in both cases we say that J1 and J2 commute because they are from different Hilbert spaces. I can't really understand what is meaning of coupling here because j1 and j2 are from different spaces so measuring one should NOT affect other, so which thing is actually coupled and why do we need two different basis(coupled and uncoupled)?

Barry
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