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If I have understood things correctly, in a multi-electron atom you have that the spin operator ##S## commutes with the orbital angular momentum operator ##L##. However, as these operators act on wavefunctions living in different Hilbert spaces, how is it possible to even calculate the commutator?

I mean, you can't calculate the commutator between ##\hat{H} = i \hbar \frac{\partial}{\partial t}## and ##\hat{p}= \frac{\hbar}{i} \vec{\nabla}## precisely because of different hiblert spaces, so why is ##L## and ##S## different?

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# Commutation between operators of different Hilbert spaces

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