Expectation value of X and Y component of angular momentum

[xinoki]

Homework Statement

Show:
<Jx>=<Jy>=0

Homework Equations

Jx=1/2(J++J-)
Jy=1/2(J+-J-)

The Attempt at a Solution

<jm l Jx l jm> = < jm l 1/2 J+ l jm> + < jm l j- l jm >
= < jm l h/2 sqrt [(j-m)(j+m+1)] + h/2sqrt[(j+m)(j+m+1) l jm >

i am not sure how to apply the next step

 

[Matterwave]

Note that |j,m> is NOT an eigenstate of either J+ or J-. In fact, J+|j,m>=k|j,m+1>, etc.

Additionally note that $$<j,m|j,n>=\delta_{m,n}$$