Angular momentum and Expectation values

[Ben4000]

Homework Statement

Express Lx in terms of the commutator of Ly and Lz and, using this result, show that <Lx>=0 for this particle.

The Attempt at a Solution

[Ly,Lz]=i(hbar)Lx

<Lx>=< l,m l Lx l l,m>

then what?

 

[Ben4000]

I can show that <Lx>=0 using the ladder opertators, but i don't think this is what is wanted from this question... how do i use
[Ly,Lz]=i(hbar)Lx to prove <Lx> = 0?

 

[gabbagabbahey]

$$\langle L_x\rangle=\langle l,m|L_x|l,m\rangle=\frac{-i}{\hbar}\langle l,m|[L_y,L_z]|l,m\rangle$$

Expand the commutator using its definition, and take the hermitian conjugate of the resulting equation...what do you see?