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\begin{document}

Clebsch gordon coefficients are used in a change of basis/representation
. 

properties for \( <j_{1}j_{2}m_{1}m_{2}|jm> \)

\begin{itemize}

\item basis independent

\item choose phase s.t. \( <j_{1}j_{2}m_{1}m_{2}|jm>= \) \( <jm|j_{1}j_{2}m_{1}m_{2}> \)

\item \( <j_{1}j_{2}m_{1}m_{2}|J_{z}|jm>= \) \( m_{1}+m_{2}=m \) (selection rule)

\item completeness \( \sum _{jm}<j_{1}j_{2}m_{1}m_{2}|jm> \)\( <jm|j_{1}j_{2}m_{1}'m_{2}'>=\delta _{m_{1}m_{1}'}\delta _{m_{2}m_{2}'} \)

\item orthogonality \( \sum _{jm}<j_{1}j_{2}jm|m_{1}m_{2}> \)\( <m_{1}m_{2}|j_{1}j_{2}j'm'>=\delta _{jj'}\delta _{m'm'} \)

\item symmetry \( <j_{1}j_{2}m_{1}m_{2}|jm>=(-1)^{j_{1}+j_{2}-j} \) \( <j_{2}j_{1}m_{1}m_{2}|jm> \) 

\item Usually defined by recursion relations

\end{itemize}

example: angular momentum coupling.

\( |j_{1}j_{2}jm>=\sum ^{j_{1}}_{m=-j_{1}}\sum ^{j_{2}}_{m_{2}=-j_{2}}|j_{1}j_{2}m_{1}m_{2}><j_{1}j_{2}m_{1}m_{2}|jm> \) where the last braket is a clebsch-gordon coefficient. Nonzero clebsch-gordon
coefficient if \( m=m_{1}+m_{2} \). Define \( n(m)= \)\# of states with fixed \( j_{1},j_{2} \) with \( J_{z}=m \). Define
\( N(j)= \)\# of 2j+1-plets with \( J^{2}=j(j+1) \). Note that \( n(m)=\sum _{j\geq m}^{\infty }N(j) \). This means \( n(m)=0 \)if \( |m|>j_{1}+j_{2} \), \( n(m)=j_{1}+j_{2}+1-|m| \) if \( j_{1}+j_{2}\geq |m|\geq |j_{1}-j_{2}| \), and
\( n(m)=2min(j_{2},j_{1})+1 \) if \( 0\leq |m|\leq |j_{1}-j_{2}| \).  This means \( N(j)=1 \) if \( |j_{1}-j_{2}|\leq j\leq j_{1}+j_{2} \) or 0 otherwise. In general, \( \sum _{j}(2j+1)N(j)=(2j_{1}+1)(2j_{2}+1)=\sum ^{j_{1}+j_{2}}_{j=|j_{1}-j_{2}|}2j+1 \). Recursion
relations: \( <j_{1}j_{2}m_{1}m_{2}|J_{\pm }|jm>=\sqrt{j(j+1)-m(m\pm 1)}<j_{1}j_{2}m_{1}m_{2}|jm\pm 1> \) \( <j_{1}j_{2}m_{1}m_{2}|J_{1\pm }+J_{2\pm }|jm>=\sqrt{j_{1}(j_{1}+1)-m_{1}(m_{1}\mp 1)}<m_{1}\mp 1m_{2}|jm> \) \( +\sqrt{j_{2}(j_{2}+1)-m_{2}(m_{2}\pm 1)}<m_{1}m_{2}\pm 1|jm> \)

\end{document}
