Coefficient of fractional parentage

In physics, coefficients of fractional parentage (cfp's) can be used to obtain anti-symmetric many-body states for like particles. In a jj-coupling scheme they are defined by the following relation.

${\mathrm{\Psi }}_{1\dots N}(j^N\alpha JM)=\sum _{{\alpha }_1{J}_1}⟨j^{N-1}({\alpha }_1{J}_1);jJ\mid \}{j}^N\alpha J⟩{[{\mathrm{\Psi }}_{1\dots N-1}(j^{N-1}{\alpha }_1{J}_1)\otimes {\psi }_N(j)]}_M^J$

The state ${\mathrm{\Psi }}_{1\dots N}(j^N\alpha JM)$ is normalized and totally anti-symmetric with respect to permutations of all its N particles, while the state ${\mathrm{\Psi }}_{1\dots N-1}(j^{N-1}{\alpha }_1{J}_1)$ is normalized and totally anti-symmetric with respect to all its N − 1 particles.