# Change coupling in a system with three angular momentum

#### d4n1el

Hi!
I'm doing some QM calculations and I'm coupling three spins j_1,j_2,j_3
If they are coupled in ls-coupling I can use the transformation
$$|j_1j_2j_3(j_{23}):JM>=\\ \sum (-1)^{j_1+j_2+j_3+j}\sqrt{2j_{12}+1}\sqrt{2j_{23}+1} \begin{Bmatrix} j_1 & j_2 & j_{12} \\ j_{3} & J & j_{23} \end{Bmatrix}|j_1j_2(j_{12})j_3:JM>$$

This is a quite standard transformation. I have founded it in a couple of places. For example in "The nuclear shell modell" of Heyde.

But now I'm more intrested in the inverse transformation, go from $$|j_1j_2(j_{12})j_3:JM>$$ to $$|j_1j_2j_3(j_{23}):JM>$$

Do you now any clever ways or identities that can be usefull. Or do I need to calculate it from scratch?

ps. sorry but i cant fix the sum-sign. It should be a summation over $$j_{12}$$

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