Expectation Values of Angular Momentum Operators

[KiwiBlack]

Homework Statement

Show that

< l,m | L_x² - L_y² | l,m > = 0

Homework Equations

L² = L_x² + L_y² + L_z²

[ L_x, L_y ] = i [STRIKE]h[/STRIKE] L_z

[ L, L_z ] = i [STRIKE]h[/STRIKE] L_x

[ L_z, L_x ] = i [STRIKE]h[/STRIKE] L_y

The Attempt at a Solution

I tried substituting different commutation values in place of L_x and L_y, 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.

 

[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 L_z 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.