Integration involving spherical harmonics

[lordkelvin]

Homework Statement

Evaluate the integral ∫∫dΩ V(Ω)Y^m_l(Ω) for V(Ω) = +V₀ for 0<θ<π/2 ; -V₀ for π/2<θ<π

Homework Equations

I was hoping to apply the orthonormality properties of the spherical harmonics but this is a little more difficult since the integral breaks into two integrals over half spheres instead of one integral over a full sphere.

I guess it may be useful to rewrite the V(Ω) as ±√(4π)Y₀⁰(Ω)V₀ in order to invoke the orthogonality properties of the spherical harmonics when integrated on a sphere.

The Attempt at a Solution

Basically I've split the integral into two integrals over the upper and lower halves of the sphere. Now what? Surely I'm not to use the definition of the spherical harmonics; there must be a cleaner way to do this by invoking the properties of Ylm(omega). There is nothing I can say about ∫∫_{half of sphere}dΩY_l^m(Y₀⁰)* as far as I know. Is there some way to combine the two integrals again so that I can use the orthonormality property?

 

[vela]

Orthonormality won't really help here because you're not integrating the product of two spherical harmonics. The integral should be pretty straightforward though. Start by integrating φ out first. Then you'll be left with an integral involving a Legendre polynomial.

 

[lordkelvin]

I get zero. I think this is so since Y_l^m(θ,$\phi$) = const* e^(i*$\phi$)*P_l^m(cos($\theta$) and so the integral over phi yields zero.

 

[vela]

It's $e^{im\phi}$, not $e^{i\phi}$. It makes a difference.

 

[paranormal]

I would suggest approaching this problem by first recognizing that the integral can be rewritten as a sum of two integrals, one over the upper half of the sphere and one over the lower half. This allows us to use the orthonormality property of the spherical harmonics, as we can now write the integral as a product of two spherical harmonics with different indices.

Next, we can use the fact that the spherical harmonics are eigenfunctions of the Laplace operator, which means that they satisfy the equation ∇^2Ylm = -l(l+1)Ylm. This can help us simplify the integrals, as we can replace the Laplace operator with its eigenvalue in the integrands.

Finally, we can use the fact that the spherical harmonics are orthogonal to each other when integrated over a full sphere, but also that they are not orthogonal when integrated over a half sphere. This means that we can use the orthonormality property of the spherical harmonics for the upper half of the sphere, but we need to introduce a negative sign for the lower half.

In summary, we can rewrite the integral as a sum of two integrals, use the eigenvalue equation for the spherical harmonics, and use the orthonormality property for the upper half of the sphere and introduce a negative sign for the lower half. This approach should allow us to evaluate the integral without having to use the definition of the spherical harmonics.