As last post discussed, finding to CG coefficient is not as straight forward as text book said by recursion.
However, there are another way around, which is by diagonalization of J^2

first we use the identity:

J^2 = J_1^2+J_2^2 + 2 J_{1z} J_{2_z} + J_{1+} J_{2-} + J_{1-} J_{2+}

when we “matrix-lize” the operator. we have 2 choice of basis. one is \left| j_1,m_1;j_2;m_2 \right> , which give you non-diagonal matrix by the J_{\pm} terms. another one is \left|j,m\right>, which give you a diagonal matrix.

Thus, we have 2 matrixs, and we can diagonalized the non-diagonal. and we have the Unitary transform P, from the 2-j basis to j basis, and that is our CG coefficient.

oh, don’t forget the normalized the Unitary matrix.

i found this one is much easy to compute.

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