# Homework Help: QM: Linear momentum of angular momentum eigenstate

1. Oct 2, 2011

### center o bass

1. The problem statement, all variables and given/known data
Find [Lz, Px] and [Lz,Py] and use this to show that $$\langle l'm'|P_x|lm\rangle = 0$$ for $$m' \neq m \pm 1.$$

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

3. 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 $$m' \neq m \pm 1?$$

2. Oct 2, 2011

### vela

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