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More specifically, what is $[S_z, p^2]$? This came up in a time-evolution problem for $\hat{S}_z(t)$, knowing that that it commutes with the non-kinetic part of some Hamiltonian $\leftrightarrow [S_z, H]=[S_z, \frac{p^2}{2m}]$

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This commutator is 0; the best way to see this is to realize that the spin part of a wave function does not have a spatial extent, and the full wave function is the product of a spatial and a spin part, each living in a different Hilbert space of states.

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