Coefficient of fractional parentage explained

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} (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} (j ^{N-1} \alpha _{1} J _{1) ⊗ \psi} ^{J} ^{M}

The state

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