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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?

Emilio Pisanty
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okj
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3 Answers3

18

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.

Community
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David
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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.

Michael Stone
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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.

Ted Bunn
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