Summation involving Clebsch–Gordan coefficients

[patric44]

Homework Statement

summation involving Clebsch–Gordan coefficients

Relevant Equations

stated below

Hi all
I am trying to follow a derivation of something involving second quantization formalism, I am stuck at this step :
$$
\sum_{m2}\sum_{\mu1}
\bra{2,m1,2,m2}\ket{k,q}\bra{2,\mu1,2,\mu2}\ket{k,-q}\delta_{-m2,\mu1}
= (-1)^{2+m2}\frac{\sqrt{2k+1}}{\sqrt{5}}\bra{k,-q,2,m2}\ket{2,-m1}\times (-1)^{2+m2}\frac{\sqrt{2k+1}}{\sqrt{5}}\bra{k,-q,2,m2}\ket{2,\mu2}
$$
i tried to use the relation:
$$
\bra{j1,m1,j2,m2}\ket{J,M} = (-1)^{2+m2}\frac{\sqrt{2J+1}}{\sqrt{2j1+1}}\;\bra{J,-M,j2m2}\ket{j1,-m1}
$$
that will reproduce the first term but not the second,
any hint on how to start

$$

 

[Unique_Engineer]

I cannot say that I remember exactly how to do this, but I remember my QM2 class then try to consult Messiah's book. (Messiah to the rescue... ).