[NOTES/QM-17003] Some Useful Restrictions on CG coefficients

In this connection with addition of angular momenta, the following results from the theory of angular momentum derived earlier will be useful. \begin{eqnarray} J_\pm\ket{JM} &=& \sqrt{(J(J+1)-M(M\pm1)}\ket{JM \pm 1}\label{E1}\\ (J_\pm^{(1)} + J_\pm^{(2)})\ket{j_1m_1j_2m_2} &=& \sqrt{(j_1(j_1+1) -(m_1\pm1)}\ket{j_1m_1\pm1j_2m_2} \nonumber\\ &+&\sqrt{(j_2(j_2+1)- m_2(m_2\pm1))}\ket{j_1m_1j_2m_2\pm1} \label{E2} \end{eqnarray} On taking conjugate of \eqref{E2} we get \begin{eqnarray} \bra{j_1m_1j_2m_2} (J_\mp^{(1)} + J_\mp^{(2)}) &=& \bra{j_1(m_1\pm1)j_2 m_2} \sqrt{(j_1(j_1+1) -(m_1\pm1)} \nonumber\\ &&\quad + \bra{j_1m_1j_2 (m_2\pm1)} \sqrt{j_2(j_2+1)- m_2(m_2\pm1))} \label{E3} \end{eqnarray} which is a consequence of the angular momentum commutation relations. Considering the matrix element \begin{equation} \matrixelement{j_1j_2m_1m_2}{J_\pm}{JM} = \matrixelement{j_1j_2m_1m_2}{(J_\pm^{(1)}+ J_\pm^{(2)})}{JM} \label{E4} \end{equation} and using \eqref{E1} and \eqref{E3} we get two relations, one for $J_+$ and \begin{eqnarray} { \sqrt{J(J+1)-M(M+1)} \innerproduct{j_1j_2m_1m_2}{J(M + 1)}} \nonumber\\ &=& \innerproduct{j_1j_2m_1-1m_2}{JM} \sqrt{j_1(j_1+1)- m_1(m_1+1)} \nonumber\\ && \quad + \innerproduct{j_1j_2m_1 m_2-1}{JM} \sqrt{j_2(j_2+1)-m_2(m_2+1)} \label{E5} \end{eqnarray} and a second relation for $J_-$ \begin{eqnarray} { \sqrt{J(J+1)-M(M-1)} \innerproduct{j_1m_1,j_2m_2}{J(M-1)}}\nonumber \\ &=& \innerproduct{j_1(m_1+1) j_2 m_2}{JM} \sqrt{j_1(j_1+1)-m_1(m_1-1)} \nonumber\\ &&\quad + \innerproduct{j_1m_1,j_2 (m_2+1)}{JM} \sqrt{j_2(j_2+1)-m_2(m_2-1)} \label{E6} \end{eqnarray} We will make repeated use of the results \eqref{E5},\eqref{E6} given above. These equations can be used successively with $M=J,J-1,\ldots$ to compute the Clebsch Gordon coefficients.