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qm-lec-17007
The allowed values of $J,M$ in the state $\ket{j_1m_1}\ket{j_2m_2}$ are restricted by the following constraints.
- The \(z\)-component of the total angular momentum, $M$, is restricted by $M=m_1+m_2$
- The total angular momentum $J^2$ can take $J(J+1)\hbar^2$, where the minimum value of $J$ is $|j_1-j_2|$ and the maximum value is $j_1+j_2$. %\label{page-resuts}
- The allowed values of $J$ are {\it all values in the range } $|j_1-j_2|$ to $j_1+j_2$ {\it in steps of unity. }
- Each combination of $J,M$ appears only once, no pair \( (J,m)\) of values of $J$ and $M$ gets repeated.
The above statements are equivalent to saying that the Clebsch Gordon coefficients are zero when any one of the above conditions is not satisfied. In addition one must remember the angular momentum results that value of $M$ must always be between $-J$ and $J$, and $J$ must be positive.
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