# Z operator and spherical harmonics

1. Feb 27, 2014

### z2394

1. The problem statement, all variables and given/known data
I want to show that <n',l',m'|$\hat{z}$|n,l,m> = 0 unless m=m', using the form of the spherical harmonics.

2. Relevant equations
Equations for spherical harmonics

3. 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!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Feb 27, 2014

### vela

Staff Emeritus
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.

3. Feb 27, 2014

### 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.