# Commutator of $\hat {L}_x$ and $\hat{V}(\hat{r})$ [duplicate]

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
$$\hat{\boldsymbol{r}}\psi=\boldsymbol{r}\psi$$
$$\hat{V}(\hat{\boldsymbol{r}})\psi=V(\boldsymbol{r})\psi$$