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.

 

[paranormal]

4.
To express a wave function in terms of spherical harmonics, you will need to use the following formula: ψ(r,θ,φ) = ΣClm Ylm(θ,φ), where Clm are coefficients and Ylm are the spherical harmonics. In this case, you will need to expand the wave function ψ(r) in terms of spherical harmonics Ylm(θ,φ) and then solve for the coefficients Clm. This can be done by using the orthogonality of spherical harmonics and integrating over the entire solid angle.

In order to expand the wave function in terms of spherical harmonics, you will need to use the following relation: Ylm(θ,φ) = (-1)^m √[(2l+1)/(4π)(l-m)!/(l+m)!] Plm(cosθ)eiφ, where Plm(cosθ) are the associated Legendre polynomials. You can then solve for the coefficients Clm by using the orthogonality relation: ∫∫Y*lm(θ,φ)ψ(r,θ,φ)r^2sinθdθdφ = Clm∫∫Y*lm(θ,φ)Ylm(θ,φ)r^2sinθdθdφ. This will give you the coefficients Clm in terms of the wave function ψ(r).

Once you have the wave function expressed in terms of spherical harmonics, you can then use the standard formulas for calculating the expectation values of L2 and Lz in terms of spherical coordinates. These can be found in most textbooks on quantum mechanics or online resources. By substituting the spherical harmonics expansion of ψ(r) into these formulas, you can then find the expectation values of L2 and Lz. I hope this helps clarify the process for you. Good luck with your calculations!