How do you apply the antisymmetrization operator?

Question

I have an expression like,

$Y^{L M_L}_{l_1 l_2}(\Omega_1, \Omega_2) = \sum_{m_1 m_2} \langle l_1 m_1l_2m_2|L m_L\rangle Y_{l_1m_1}(\Omega_1) Y_{l_2m_2}(\Omega_2)$ ,

as the angular part of a two electron wavefunction containing a Clebsch-Gordan coefficient and spherical harmonics. I wish to apply the antisymmetrization operator and invoke the properties of the Clebsch-Gordan coefficient. I have read that the antisymmetrization operator interchanges the coordinates or labels, but cannot find in any literature the explicit application of the antisymmetrization operator on such an expression. What should the expression above look like when $A_{12}$ is applied to both sides. Do I interchange every label containing the 1 and 2 ($l_1, m_1$, $\Omega_1$ and $\Omega_2$), or only the coordinates (or only the subscript labels)? An explanation of how it should be applied would be great.

Thanks.

Answer 1

Antisymmetrization is usually done w/r to coordinates so that $$ \Phi(r_1)\Psi(r_2)-\Phi(r_2)\Psi(r_1) $$ is antisymmetric. In your case $\Phi\to Y_{\ell_1m_1}$ and $\Psi\to Y_{\ell_2m_2}$.

This can be done systematically using Young’s symmetrizers; normalization is done at the end.