# How to prove spherical harmonics are orthogonal

A lot of texts and derivations eg here simply say:

"The Spherical Harmonics are orthonormal, so:

$$\int{ Y_l^m Y_{l'}^{m'} } = \delta_{ll'}\delta_{mm'}$$

And if you try any (l,m) pair you will find this always works out.

But how do you prove they are orthonormal for every $l$ and $m$? Where do you start? What principles do you use?

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Maybe not really an answer but you may get the idea nontheless: this is true more or less by construction. You get the spherical harmonics (as an example) as eigenfunctions of the angular part of the Laplace Operator, that is, they satisfy $$\Delta_{S^2} Y_{lm}(\vartheta,\phi) = \lambda Y_{lm} (\vartheta,\phi)$$ (Actually it turns out that this implies $\lambda = -l(l+1)$ with integer $l$) If you have such eigenfunctions for different eigenvalues it is a matter of linear algebra to show they are orthogonal, by looking at $$\int_{S^2}\langle \nabla_{S^2}Y_{lm}, \nabla_{S^2}Y_{l'm'}\rangle d\mu_{S^2}= -\int_{S^2}\langle Y_{lm}, \Delta_{S^2}Y_{l'm'}\rangle d\mu_{S^2}$$ This implies that the functions are orthogonal if $l\neq l'$, since otherwise you could derive $l(l+1) = l'(l'+1)$ from this. For fixed $l$ it turns out that you may solve the equation by a separation approach which leads to an ODE which is known to be solvable by orthogonal polynomials by ODE theory.
You can also write down the $Y_{ml}$ quite explicitly, see e.g. the german wikipedia page on "Kugelflächenfunktionen" http://de.wikipedia.org/wiki/Kugelfl%C3%A4chenfunktionen. If you look at these more closely and and do have some ODE background you may notice that the $\vartheta$ part are well known orthogonal polynomials in $\cos(\vartheta)$ (Legendre Polynomials), while the $\phi$ part is more or less just $e^{im\phi}$ which is known to a system of orthogonal functions. To then actually prove orthogonality is still a bit of work, but it kind of shows you the direction you should take.
Neat answer. I was not aware it was by construction! So there isn't a way to write an explicit proof, starting with $Y_l^m$. – bobobobo Jun 4 '12 at 14:49