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
[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 interested 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 can't fix the sum-sign. It should be a summation over [tex]j_{12}[/tex]

 

[eljose79]

and j_{23}.

Hello!

Thank you for your question. The inverse transformation from |j_1j_2(j_{12})j_3:JM> to |j_1j_2j_3(j_{23}):JM> can be derived using the Clebsch-Gordan coefficients. These coefficients are used to couple angular momenta in quantum mechanics and can be found in many textbooks, including the one you mentioned by Heyde.

To derive the inverse transformation, you can use the following formula:

|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>

First, you need to use the fact that the Clebsch-Gordan coefficients are symmetric under the exchange of the two spins j_1 and j_2. This means that the coefficients for j_1 and j_2 can be interchanged without changing the result. Using this, you can rewrite the formula as:

|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_2 & j_1 & j_{12} \\
j_{3} & J & j_{23}
\end{Bmatrix}|j_2j_1(j_{12})j_3:JM>

Next, you can use the fact that the Clebsch-Gordan coefficients are also symmetric under the exchange of j_3 and j_{12}. This means that the coefficients for j_3 and j_{12} can be interchanged without changing the result. Using this, you can rewrite the formula as:

|j_1j_2j_3(j_{23}):JM>=\\
\sum
(-1)^{j_1+j_2+j_3+j}\sqrt{2j_{12}+