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.
Ψ 1 … N ( j N α J M ) = ∑ α 1 J 1 ⟨ j N − 1 ( α 1 J 1 ) ; j J ∣ } j N α J ⟩ [ Ψ 1 … N − 1 ( j N − 1 α 1 J 1 ) ⊗ ψ N ( j ) ] M J {\displaystyle \Psi _{1\ldots N}(j^{N}\alpha JM)=\sum _{\alpha _{1}J_{1}}\langle j^{N-1}(\alpha _{1}J_{1});jJ\mid \}j^{N}\alpha J\rangle \left[\Psi _{1\ldots N-1}(j^{N-1}\alpha _{1}J_{1})\otimes \psi _{N}(j)\right]_{M}^{J}}
The state Ψ 1 … N ( j N α J M ) {\displaystyle \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 Ψ 1 … N − 1 ( j N − 1 α 1 J 1 ) {\displaystyle \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.
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