$J$-pairing interaction, number of states, and nine-$j$ sum rules of four identical particles

Abstract

In this paper we study $J$-pairing Hamiltonian and find that the sum of eigenvalues of spin $I$ states equals sum of norm matrix elements within the pair basis for four identical particles such as four fermions in a single-$j$ shell or four bosons with spin $l$. We relate number of states to sum rules of nine-$j$ coefficients. We obtained sum rules for nine-$j$ coefficients $<(jj)J,(jj)K:I|(jj)J,(jj)K:I>$ and $<(ll)J,(ll)K:I|(ll)J,(ll)K:I>$ summing over (1) even $J$ and $K$, (2) even $J$ and odd $K$, (3) odd $J$ and odd $K$, and (4) both even and odd $J,K$, where $j$ is a half integer and $l$ is an integer.

arXiv comment6 pages, no figure, updated version, to be published. Physical Review C, in press

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Cited by in corpus (8)

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• Number of states for nucleons in a single-$j$ shell
• Special 6j and 9j symbols for a single-j shell
• Total number of $J$ levels for identical particles in a single-$j$ shell using coefficients of fractional parentage
• Zeros of 6j Symbols:Atoms:Nuclei and Bosons
• More on Asymptotic behaviors of 9j Coefficients
• Number of spin-J states and odd-even staggering for identical particles in a single-j shell