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I know that the Clebsch-Gordan coefficient, $$\left<J_1, m_1, J_2, m_2|J,M\right>,$$ is zero if the following conditions are not satisfied: $$|J_1-J_2| \le J \le J_1+J_2,$$ $$m_1+m_2=M,$$ $$|M| \le J.$$ My question is whether it is possible that a Clebsch-Gordan coefficient satisfies these conditions but is still zero? I.e. are these conditions been satisfied a necessary and sufficient condition for a non-zero Clebsch-Gordan coefficient or only a necessary? and can it be proved either way?

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A Clebsch-Gordan coefficient can be zero even if those conditions are satisfied. For example*, using the notations $|J, M\rangle$ and $|J_1, m_1, \, J_2, m_2 \rangle$: $$ |\,2, 0\,\rangle= \frac{1}{\sqrt{2}}|\,2,+1,\,1,-1\,\rangle - \frac{1}{\sqrt{2}}|\,2,-1,\,1,+1\,\rangle \,, $$ so that there is no projection on the $m_1=m_2=0$ state: $$ \langle J, M |J_1, m_1,\, J_2, m_2 \rangle = \langle \,2,0\,| \,2,0,\,1,0\, \rangle = 0\,. $$

*I basically searched for a zero entry in Table 4.7 of David J. Griffiths' Introduction to Quantum Mechanics, 2nd edition.

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The topic of "accidental" zeros of Clebsh-Gordon coefficients is still active. See this paper as an example of efforts to classify these non-trivial zeros.

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