There is an identity concerning spherical harmonics that plays a pretty important role in atomic physics. Thanks to wikipedia (http://en.wikipedia.org/wiki/Spherical_harmonic) I know that its name is Unsöld's Theorem. It states that:

$$ \sum_{m = -l}^{l}|Y_{l}^{m}(\phi,\theta)|^2 = \frac{2l+1}{4 \pi} $$

However I was not able to find any proof over the Internet, nor solve it myself. I would be grateful for any tips, heuristics, cause I have no idea, where to even begin with.


1 Answer 1


This is an easy consequence of a more general addition formula for spherical harmonics: \begin{align}P_l\left(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\cos(\varphi_1-\varphi_2)\right)=\\ =\frac{4\pi}{2l+1}\sum_{m=-l}^l Y_l^m(\theta_1,\varphi_1)\overline{Y_l^m(\theta_2,\varphi_2)}.\tag{1} \end{align} Namely, set $\theta_1=\theta_2=\theta$, $\varphi_1=\varphi_2=\varphi$ and use the well-known normalization $P_l(1)=1$ of the Legendre polynomials. For the proof of (1), see Whittaker-Watson, Subsection 18.4, p. 395.

The basic idea is that the argument of the Legendre polynomial on the left can be represented as a scalar product of two vectors corresponding to the points $(\theta_1,\varphi_1)$ and $(\theta_2,\varphi_2)$ on the unit sphere. Hence the left side belongs to the eigenspace of the square of angular momentum with orbital number $l$ with respect to both sets of spherical coordinates.


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