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.



\Psi_{1\ldots N}(j^N\alpha JM)= \sum_{\alpha_1 J_1} \langle j^{N-1}(\alpha_1 J_1);j J\mid\} j^N \alpha J \rangle \left[\Psi_{1\ldots N-1}(j^{N-1}\alpha_1 J_1)\otimes \psi_N(j)\right]^J_M $$

The state $$\Psi_{1\ldots N}(j^N\alpha JM)$$ is normalized and totally anti-symmetric with respect to permutations of all its $N$ particles, while the state $$ \Psi_{1\ldots N-1}(j^{N-1}\alpha_1 J_1) $$ is normalized and totally anti-symmetric with respect to all its $N − 1$ particles.