Final remarks

Note finally, that the number of basis vectors is generally given by $(2j_1+1)(2j_2+1)$, and that this number is equal to

$$\sum_{J=\vert j_1-j_2\vert}^{j_1+j_2}(2J+1) = (2j_1+1)(2j_2+1)$$

Also, sometimes one sees the so called ``$3J$'' symbols used instead of Clebsch-Gordan coefficients. This are denoted as

$$\left(\matrix{j_1 & j_2 & J \cr m_1 & m_1 & M}\right)$$

and are related to the Clebsch-Gordan coefficients by

$$\left(\matrix{j_1 & j_2 & J \cr m_1 & m_1 & M}\right) = (-1... ...\sqrt{2J+1}{\langle j_1\;\;m_1;j_2\;\;m_2\vert}{J\;\;M\rangle}$$