Homework Help: Express wave function in spherical harmonics

1. Jan 27, 2014

z2394

1. Problem:
I have a wave function ψ(r) = (x + y + z)*f(r) and want to find the expectation values of L2 and Lz. It is suggested that I first change the wave function to spherical coordinates, then put that in terms of spherical harmonics of the form Yl,m.

2. Relevant equations:
Spherical harmonics and conversions from Cartesian to spherical coordinates

3. Attempt at solution:
So I know how to express the wave function in spherical coordinates (which should be ψ(r) = r*f(r)*(sinΘcosø + sinΘsinø + cosΘ). I am having a hard time going from this to spherical harmonics though (I am sort of new to this). I can see from a table of spherical harmonics that Y1,0 does have a cosΘ, but how would I get the terms that have Θ and ø? (I can see that there are some spherical harmonics that have e, but this would give me cosø + isinø, so it doesn't look like that would get me what I want.)

2. Jan 28, 2014

CAF123

Hi z2394,
The wavefunction can be written rf(r)sinθ[cosø + sinø] + rf(r)cosθ. As you said, the second term here can be written in terms of a single spherical harmonic. To get the first term as a linear combination of spherical harmonics, try expressing cosø and sinø in their complex exponential forms.

3. Jan 28, 2014

z2394

So I see how to do that now, and I get ψ(r) = r*f(r)*($\sqrt{2{\pi}/3}($-Y1,1+Y1,-1) - (1/i)$\sqrt{2{\pi}/3}$(Y1,1 + Y1,-1) + $\sqrt{4{\pi}/3}$Y1,0). But now how do I use this to get <L2> and <Lz>? I know to do <ψ| L2 |ψ> and <ψ| Lz |ψ>, but does L2 acting on ψ here still give me the l(l+1)h2 eigenvalue, and Lz the mh eigenvalue, (where h is actually the reduced Planck constant), when acting on this ψ?

4. Jan 29, 2014

CAF123

When L2 and Lz act on the spherical harmonics, then the eigenvalue returned is l(l+1)h2 and mh respectively. First though, you will need to normalize ψ. Introduce some normalization constant, say A, and solve for A using the fact that the spherical harmonics are a complete set of orthonormal functions.