QM: Linear momentum of angular momentum eigenstate

[center o bass]

Homework Statement

Find [Lz, Px] and [Lz,Py] and use this to show that $$\langle l'm'|P_x|lm\rangle = 0$$ for $$m' \neq m \pm 1.$$

Homework Equations

$$L_z|lm\rangle = \hbar m |lm\rangle$$
$$L^2|lm\rangle = \hbar^2 l(l+1)|lm\rangle$$
$$L_{\pm}|lm\rangle = \hbar \sqrt{l(l+1)-m(m\pm 1)}|l,m\pm 1 \rangle$$

The Attempt at a Solution

It was easy to show that $$[L_z,P_x] = i \hbar P_y$$ and that $$[L_z,P_y] = - i\hbar P_x$$ but how might I use this to show that $$\langle l'm'|P_x|lm\rangle = 0$$ for $$m' \neq m \pm 1?$$

 

[vela]

Start with $-i\hbar\langle l' m' \lvert P_x \rvert l m \rangle = \langle l' m' \lvert [L_z,P_y] \rvert l m \rangle$.