- #1

- 289

- 39

- 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

$$

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

$$