Change coupling in a system with three angular momentum

  • Thread starter d4n1el
  • Start date
3
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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
[tex]|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>
[/tex]

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 [tex]|j_1j_2(j_{12})j_3:JM>[/tex] to [tex]|j_1j_2j_3(j_{23}):JM>[/tex]

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 [tex]j_{12}[/tex]
 

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