Consider the angular momentum operator $\hat{L_x}=\hat y\hat{p}_z-\hat{z}\hat{p}_y$ and the potential operator $\hat{V}$ where the potential $\hat{V}=\hat{V}(\hat{r})$ is spherically symmetric.

It can be shown that $[\hat {L}_x, \hat {V}] = \hat y [\hat {p}_z, \hat V]-\hat z [\hat {p}_y, \hat V]. $ I tried showing this myself and found that this is only true if $$[\hat y, \hat V]=0 $$ and $$[\hat z, \hat V]=0. $$

Why are the last two equations equal to zero?


Because the operator $\hat{\boldsymbol{r}}$ is just


and similarly


i.e. simply multiplication with a real number. And as you know, real numbers are commutative.

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