\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\oddsidemargin}{0in}
\setlength{\evensidemargin}{0in}
\setlength{\textwidth}{6.5in}
\setlength{\parindent}{0in}
\setlength{\parskip}{.1in}
\setlength{\topmargin}{0in}
\setlength{\headsep}{0in}
\setlength{\headheight}{0in}
\setlength{\textheight}{9in}



\begin{document}
 
\begin{flushright}
\today
\end{flushright}
\begin{center}
\bf{PHY 742 -- Notes on Clebsch-Gordan coefficients}
\end{center}
\raggedright

\bigskip
Reference: Abramowitz \& Stegum -- pg. 1006

Original formula:
\begin{eqnarray}
&<j_1 j_2 m_1 m_2|j_1 j_2 j m>=  \delta_{m,m_1+m_2} \sqrt{\frac{(j_1+j_2-j)!(j_1-j_2+j)!
(-j_1+j_2+j)!(2j+1)}{(j_1+j_2+j+1)!}}\\ \nonumber
&\times \sum_k \frac { (-1)^k \sqrt{(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!(j+m)!(j-m)!}}{k!(j_1+j_2-j-k)!(j_1-m_1-k)!(j_2+m_2-k)!(j-j_2+m_1+k)!(j-j_1-m_2+k)!}
\end{eqnarray}

Working formula:
\begin{eqnarray}
&<j_1 j_2 m_1 m_2|j_1 j_2 j m>=  \delta_{m,m_1+m_2} \sqrt{\frac{(j_1-j_2+j)!
(-j_1+j_2+j)!(2j+1)}{(j_1+j_2-j)!(j_1+j_2+j+1)!}}\\ \nonumber
&\times \sqrt{\frac{(j_1+m1)!(j_2-m_2)!(j+m)!}{(j_1-m1)!(j_2+m_2)!(j-m)!}}
\\ \nonumber
&\times \sum_k \frac {(-1)^k }{k!} \frac{(j_1+j_2-j)!}{(j_1+j_2-j-k)!}\frac{(j_1-m_1)!}{(j_1-m_1-k)!}\frac{(j_2+m_2)!}{(j_2+m_2-k)!}\frac{(j-m)!}{(j-j_2+m_1+k)!(j-j_1-m_2+k)!} 
\end{eqnarray}

Additional reference for Clebsch-Gordan coefficients:  Complement
{\bf{B_X}} and {\bf{C_X}} of Cohen-Tannoudji's text (Volume \#2). 

A related quantity is the Gaunt coefficient:
\begin{displaymath}
G^{L M}_{\ell_1m_1 \ell_2m_2} \equiv \sqrt{4 \pi}
\int d\Omega Y^*_{\ell_1m_1}({\bf{\hat{r}}})Y^*_{LM}({\bf{\hat{r}}})
Y_{\ell_2m_2}({\bf{\hat{r}}}).
\end{displaymath}
As discussed in Cohen-Tannoudji's text, the Gaunt coefficent can be expressed
in terms of a product of two Clebsch-Gordan coefficients and hence
can also be calculated analytically. 



\end{document}
