Why Does <n',l',m'|\hat{z}|n,l,m> Equal Zero Unless m=m'?

[z2394]

Homework Statement

I want to show that <n',l',m'|$\hat{z}$|n,l,m> = 0 unless m=m', using the form of the spherical harmonics.

Homework Equations

Equations for spherical harmonics

The Attempt at a Solution

Not sure how to begin here since there aren't any simple eigenvalues for $\hat{z}$|n,l,m>. I have a feeling that it may have something to do with normalization of the spherical harmonics (because they have Legendre polynomials that are P(cosΘ) = P(z) and would also give you a exp(imø)*exp(im'ø) term), but I have no idea how this could actually give you something for $\hat{z}$as an operator, or something you could actually use to figure out $\hat{z}$|n,l,m>.

Any help at all would be appreciated!

 

[vela]

You essentially have the answer already. In the coordinate basis, the operator $\hat{z}$ is represented by $r\cos\theta$. Just write down the integral for the inner product and evaluate it.

 

[z2394]

Thanks! I guess I was thinking about it in an operator sense, so it had not occurred to me to do it as an integral instead.