What are the uses of spherical harmonics?

[member 428835]

Hi PF!

When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of $\phi,\theta$ but not $r$. Why don't they include the $r$ component?

Thanks!

 

[Orodruin]

Hi PF!

When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of $\phi,\theta$ but not $r$. Why don't they include the $r$ component?

Thanks!

Because they are functions of $\theta$ and $\phi$ by definition. They are functions defined on a sphere and are the eigenfunctions of the angular part of the Laplace operator.

Of course, if you want to make an expansion of a function of $r$, $\theta$, and $\phi$ in terms of spherical harmonics, then the expansion coefficients will depend on $r$. This is just separation of variabels and should be well described in any textbook on the subject.

 

[Vanadium 50]

Why don't they include the rr component?

Because they are functions of θ\theta and ϕ\phi by definition.

Right. They are useful because they allow you to separate out the angular and radial parts of the problem.

 

[Orodruin]

Right. They are useful because they allow you to separate out the angular and radial parts of the problem.

In this particular instance, yes. Generally they have other uses as well - such as series expanding any function defined on the surface of a sphere (eg, the CMB temperature variations).