Clebsch-Gordan coefficients

The Clebsch-Gordan coeffficients \( \langle lm_lsm_s|jm_j\rangle \) have some interesting properties for us, like the following orthogonality relations $$ \sum_{m_1m_2}\langle j_1m_1j_2m_2|JM\rangle\langle j_1m_1j_2m_2|J'M'\rangle=\delta_{J,J'}\delta_{M,M'}, $$ $$ \sum_{JM}\langle j_1m_1j_2m_2|JM\rangle\langle j_1m_1'j_2m_2'|JM\rangle=\delta_{m_1,m_1'}\delta_{m_2,m_2'}, $$ $$ \langle j_1m_1j_2m_2|JM\rangle=(-1)^{j_1+j_2-J}\langle j_2m_2j_1m_1|JM\rangle, $$ and many others. The latter will turn extremely useful when we are going to define two-body states and interactions in a coupled basis.