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## Homework Statement

Find the Clebsch Gordon coefficients of [tex] 1/2 \otimes 1 = 3/2 \oplus 1/2 [/tex]

## Homework Equations

We have some properties of the CG coefficients which might be useful:

1) they are nonzero only if j is between j1-j2 and j1+j2

2) m = m1+m2 for nonzero coefficients

3) they are real

## The Attempt at a Solution

I am horribly confused. I know that the CG coefficients are given as the coefficients in the expansion

[tex] |j m, j_1 j_2 \rangle = \sum_{m_1} \sum_{m_2} | j_1 m_1, j_2 m_2 \rangle \langle j_1 m_1, j_2 m_2| jm, j_1 j_2\rangle [/tex]

or

[tex] \langle j_1 m_1, j_2 m_2 | j m \rangle [/tex]

and I know that the possible |j m j1 j2> states are for the product space with j1 = 1, j2 = 1/2:

[tex] | \frac{3}{2} \frac{3}{2} ,1 \frac{1}{2} \rangle, | \frac{3}{2} \frac{1}{2},1 \frac{1}{2} \rangle, | \frac{3}{2} \frac{-1}{2} ,1 \frac{1}{2} \rangle, | \frac{3}{2}\frac{-3}{2} ,1 \frac{1}{2} \rangle, | \frac{1}{2} \frac{1}{2} ,1 \frac{1}{2} \rangle, | \frac{1}{2} \frac{-1}{2} ,1 \frac{1}{2} \rangle [/tex]

but I don't understand what on earth to do or where to even start. Any help would be great.