How to show that $$\langle\vec{r}|[L_{j},f(\hat{r})]|\psi\rangle = 0~?$$
note that $\hat{r}$ is a operator and not a unit vector What I know so far is that the commutation relation of a normal radial function the angular moment $[L_{j}, f(r)] = 0$ as well as that radial function of a operator $\hat{r}$ by $$ f(\hat{r}) = \int d^{3} r' f(r') |\vec{r'}\rangle \langle\vec{r'}| $$ Based on this, one can imply the eigenvalue problem $f(\hat{r})|\vec{r}\rangle = f(r)|\vec{r}\rangle $ . Another thing I know is that $\langle\vec{r}|p_{k}|\psi\rangle = \frac{\hbar}{i}\frac{d}{dx_{k}} \langle\vec{r}|\psi\rangle$ and $L_{j} = \epsilon_{jkm}x_{k}p_{m}$. I tried expanding out the commutation in the middle and then split the expression into $\langle\hat{r}|L_{j}f(\hat{r})|\psi\rangle - \langle\vec{r}|f(\hat{r})L_{j}|\psi\rangle$ I was thinking next of moving the operators around and evaluating what they work on?