# Expectation Values of Angular Momentum Operators

1. Sep 4, 2012

### KiwiBlack

1. The problem statement, all variables and given/known data
Show that

< l,m | Lx2 - Ly2 | l,m > = 0

2. Relevant equations

L2 = Lx2 + Ly2 + Lz2

[ Lx, Ly ] = i [STRIKE]h[/STRIKE] Lz

[ L, Lz ] = i [STRIKE]h[/STRIKE] Lx

[ Lz, Lx ] = i [STRIKE]h[/STRIKE] Ly

3. The attempt at a solution

I tried substituting different commutation values in place of Lx and Ly, but I'm not reducing it any further. I also tried ladder operations, but my professor said they're not needed to solve the problem.

2. Sep 4, 2012

### Oxvillian

Well I think ladder operators are fine, but there are some "shortcut" ways too:

You could probably get away with just saying that it's "obvious" that
$$\langle l,m \lvert L_x^2 \lvert l,m \rangle = \langle l,m \lvert L_y^2 \lvert l,m \rangle$$
because an Lz eigenstate shouldn't know the difference between the x and y directions. If you wanted to make that idea precise, you could find out what happens to the eigenstate and to the angular momentum operators when you do a 90-degree rotation about the z-axis.