Express wave function in spherical harmonics

[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 L_z. It is suggested that I first change the wave function to spherical coordinates, then put that in terms of spherical harmonics of the form Y_l,m.

2. Homework 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 Y_1,0 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^iø, but this would give me cosø + isinø, so it doesn't look like that would get me what I want.)

 

[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.

 

[z2394]

So I see how to do that now, and I get ψ(r) = r*f(r)*(\sqrt{2{\pi}/3}(-Y_1,1+Y_1,-1) - (1/i)\sqrt{2{\pi}/3}(Y_1,1 + Y_1,-1) + \sqrt{4{\pi}/3}Y_1,0). But now how do I use this to get <L²> and <L_z>? I know to do <ψ| L² |ψ> and <ψ| L_z |ψ>, but does L² acting on ψ here still give me the l(l+1)h² eigenvalue, and L_z the mh eigenvalue, (where h is actually the reduced Planck constant), when acting on this ψ?

 

[CAF123]

I know to do <ψ| L² |ψ> and <ψ| L_z |ψ>, but does L² acting on ψ here still give me the l(l+1)h² eigenvalue, and L_z the mh eigenvalue, (where h is actually the reduced Planck constant), when acting on this ψ?

When L² and L_z 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.