Expectations and Uncertainties of Lx^2 and Ly^2 in Eigenstate |l,m>

[azone]

Homework Statement

Show that for the eigenstate |l,m> of L^2 and Lz, the expectation values of Lx^2 and Ly^2 are <Lx^2>=<Ly^2>=1/2*[l(l+1)-m^2]hbar^2

and for uncertainties, show that deltaLx=deltaLy={1/2*[l(l+1)-m^2]hbar^2}^(0.5)

Homework Equations

eigenvalues of L^2 are l(l+1)hbar^@
eigenvalues of Lz are m*hbar

The Attempt at a Solution

I noticed that the expectation values are very closely related to the eigenvalues of L^2 and Lz. So I tried using commutator relationships to somehow get Lx^2 as a result.
[Lz,Lx^2] = ihbar(Lx*Ly+Ly*Lx)
[Lx,Ly^2] = -ihbar(Lx*Ly+Ly*Lx)
[L^2,Lx] = [L^2,Ly] = 0
but none of these seem to help at all...any suggestions on how to approach the problem?

 

[Physics Monkey]

Two hints for you azone:

1) $$L^2 = L_z^2+L_x^2 +L_y^2 [\tex]<br /> <br /> 2) There is symmetry between x and y$$