Coefficient of Fractional Parentage¶
Resolve an antisymmetric many-fermion state into coupled one-particle-plus-parent channels using convention-dependent expansion amplitudes.
Core Idea¶
A coefficient of fractional parentage (CFP) is an expansion amplitude connecting a fully antisymmetric \(N\)-fermion state to a channel made from an antisymmetric \((N-1)\)-particle parent coupled with one more particle. The coupled component need not itself be antisymmetric under exchange of the added particle; the appropriately weighted sum represents the finished target state. Coefficient values depend on basis, coupling, labels, normalization and phase convention, and do not literally describe physical ancestry.[^ref-fba11b269744]
Scope of Application¶
Atomic open-shell electron and nuclear shell-model calculations use parentage decompositions to organize many-particle states and operator matrix elements. In Deveikis and Kamuntavičius's \(j=5/2\), \(N=4,J=6,T=0\) worked case, five parent-plus-particle channels yield an antisymmetrizer matrix of rank two; their \(4(2)\) state has first coefficient \(5/(2\sqrt{15})=\sqrt{A_{11}}\) for \(A_{11}=5/12\), and all five reported coefficients square to one in sum. A tabulation applies only to its shell, coupled basis and phase convention; related reduced coefficients have their own rules. This entry is limited to antisymmetric fermion states, not bosonic analogues.[ref-55c6b62e4e09][ref-fba11b269744]
Clarity¶
The “parent” is a lower-particle basis state, not a progenitor. A Clebsch–Gordan coefficient couples angular momenta, but a CFP additionally identifies a component of a symmetry-constrained many-body target. The antisymmetrizer may be used to calculate these amplitudes; it is not itself a CFP.[^ref-fba11b269744]
Manages Complexity¶
Instead of repeatedly handling the full \(N\)-particle antisymmetric state directly, one uses smaller parent sectors, couples in a particle and reuses the resulting amplitudes. Degenerate states may require extra distinguishing labels, while mismatched phase conventions can make two correct tables look inconsistent.[ref-fba11b269744][ref-55c6b62e4e09]
Abstract Reasoning¶
Fix the identical-particle sector, shell, target state, complete quantum-number labels, coupling order and phase convention. Enumerate compatible antisymmetric parents, couple each to one particle, and determine the amplitudes that reconstruct the fully antisymmetric target. Validate normalization and convention before using the coefficients in a matrix-element calculation.[^ref-fba11b269744]
Knowledge Transfer¶
Atomic electrons and modeled nuclear particles differ physically, yet both use the same parent-plus-particle decomposition strategy. The strategy transfers across settings; the numerical coefficient array does not transfer without matching basis and conventions. Coefficient is the formal genus of the expansion amplitude. Nuclear Shell Model is an application and Antisymmetrizer a related operator, not its strict parent.
[^ref-fba11b269744]: Deveikis and Kamuntavičius, “The Coefficients of Fractional Parentage of Nuclear Shell Model,” original research preprint. Definition, parent expansion, antisymmetry and calculation methods. [^ref-55c6b62e4e09]: Gaigalas and Fritzsche, “Calculation of Reduced Coefficients and Matrix Elements in \(jj\)-Coupling,” original research preprint. Atomic open-shell CFPs, reduced coefficients and matrix-element calculations.
Relationships to Other Abstractions¶
Current abstraction Coefficient of Fractional Parentage Domain-specific
Parents (1) — more general patterns this builds on
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Coefficient of Fractional Parentage is a kind of Coefficient Domain-specific
A fractional-parentage amplitude is a coefficient in a specified state expansion.
Hierarchy path (1) — routes to 1 parentless root
- Coefficient of Fractional Parentage → Coefficient → Representation → Abstraction
Neighborhood in Abstraction Space¶
Coefficient of Fractional Parentage sits in a sparse region of the domain-specific corpus (64th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Statistical Mechanics & Particle Phenomena (15 abstractions)
Nearest neighbors
- Slater Determinant — 0.86
- Pseudo-Jahn–Teller Effect — 0.85
- Random-Phase Approximation — 0.85
- Mass–Energy Equivalence — 0.84
- Stimulated Raman Adiabatic Passage — 0.83
Computed from structural-signature embeddings · 2026-10-08