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Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?:

\begin{align}\int_0^{2\pi}\int_0^\pi Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi)&Y_{l_3}^{m_3}(\theta,\phi)\sin(\theta)d\theta d\phi =\\ &\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \left( {\begin{array}{ccc} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \\ \end{array} } \right) \left( {\begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \\ \end{array} } \right) \end{align}

Where the $Y_{l}^{m}(\theta,\phi)$ are spherical harmonics. Or does anyone know of a reference where the derivation is given?

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    $\begingroup$ May that link would be of use? physics.stackexchange.com/q/4789 $\endgroup$
    – Kostya
    Commented May 18, 2011 at 20:54
  • $\begingroup$ Arfken, Mathematical Methods for Physicists, 3rd ed., 1985, p.700 $\endgroup$
    – Ramashalanka
    Commented May 18, 2011 at 21:11
  • $\begingroup$ @Ramashalanka: Arfken mentions the identity but doesn't provide a derivation, what I'm interested in is the actual derivation. $\endgroup$
    – okj
    Commented May 18, 2011 at 22:00
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    $\begingroup$ As Sakurai also pointed out in his book, this is an application of the Wigner-Eckart Theorem. $\endgroup$
    – wilsonw
    Commented Feb 27, 2017 at 13:01

3 Answers 3

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Sakurai, Modern Quantum Mechanics, 2nd Ed. p.216

In his derivation the product of the first two spherical harmonics is expanded using the Clebsch-Gordan Series (which is also proved) to get the following equation.

$Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi)\ =$

$\displaystyle\sum\limits_{l} \displaystyle\sum\limits_{m} \sqrt{\frac{(2l_1+1)(2l_2+1)(2l+1)}{4\pi}} \left( {\begin{array}{ccc} l_1 & l_2 & l \\ 0 & 0 & 0 \\ \end{array} } \right) \left( {\begin{array}{ccc} l_1 & l_2 & l \\ m_1 & m_2 & -m \\ \end{array} } \right)(-1)^m Y_{l}^{m}(\theta,\phi)$

Which makes the integral much easier.

Final Note: Sakurai writes his derivation in Clebsch-Gordan coefficients so the equation was changed to fit with the question asked.

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It's a special case of the integral for three representation matrices that is derived in Wigner's article on page 91 of Biedenharn and van Dam "quantum theory of angular momentum" (wigner's equation 5). You need to know the connection between the representation matrices $D^j_{mn}(U)$ and the $Y^l_m$'s. You can find that explained in Chapter 15 of Stone and Goldbart. It's on page 620 in the online version at http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf.

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  • $\begingroup$ Thank you for making your excellent book available to the world free of charge. $\endgroup$
    – Amey Joshi
    Commented Jan 9, 2021 at 12:48
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Richard Zare's book has a derivation, or at least all the pieces of one. Section 3.9 has the key results, although you have to refer back to previous sections to complete the derivation.

I wonder if you could also do it by induction, using the recurrence relations for the spherical harmonics on the left and for the 3-$j$'s on the right. I've never seen anyone do anything like this, maybe it doesn't work.

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