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I'm reading a bit about Clebsch-Gordan coefficients and I found two things in their general description I didn't quite understand. Can anyone help me with these questions?

First, I read that the Clebsch-Gordan coefficients are zero unless the total angular momentum satisfies [itex]|j_1-j_2|\leq j \leq j_1+j_2[/itex]. How would you prove this?

Second, and this is a bit more difficult, I know the CG coefficients are elements of a change of basis matrix, which relates the two bases

1. [itex]|j_1 m_1 \rangle \otimes |j_2 m_2 \rangle[/itex]

2. [itex]|j_1j_2jm \rangle[/itex].

I also know that with respect to the basis 1., the total angular momentum operator [itex]J_i[/itex] is represented by the matrix [itex]J_{1i} \otimes \mathbb{1} + \mathbb{1} \otimes J_{2i}[/itex]. Now I wish to know how this matrix looks with respect to the other basis, perhaps using the change of basis matrix.

I know this is the 'tensor product decomposition' rule, but I don't fully understand what's going on. How does that matrix look in the other basis?

Thanks in advance

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# Two questions on Clebsch-Gordan coefficients

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