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Journal of the Physical Society of Japan

Generalized Seniority Scheme

Akito Arima, Hideo Kawarada

Published Oct 1, 1964• 24 citations
Publisher

Abstract

The seniority scheme introduced by Racah is generalized in the case of degenerate levels. Many formulae for the pure configuration j n are modified in order to calculate matrix elements in this generalized seniority scheme. It is especially the most interesting result that the diagonal matrix element of two-body operators between the states with the same seniority v can be expressed in terms of the matrix elements in the v -particle system. This result, of course, holds in the case of the pure configuration.

DOI
10.1143/jpsj.19.1768

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arXiv:1010.2415•AIP conference proceedings•Jan 1, 2010•4 citations
The seniority scheme introduced by Racah is generalized in the case of degenerate levels. Many formulae for the pure configuration j n are modified in order to calculate matrix elements in this generalized seniority scheme. It is especially the most interesting result that the diagonal matrix element of two-body operators between the states with the same seniority v can be expressed in terms of the matrix elements in the v -particle system. This result, of course, holds in the case of the pure configuration.
A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
The use of the seniority quantum number in many‐body systems is reviewed. A brief summary is given of its introduction by Racah in the context of atomic spectroscopy. Several extensions of Racah’s original idea are discussed: seniority for identical nucleons in a single‐j shell, its extension to the case of many, non‐degenerate j shells and to systems with neutrons and protons. To illustrate its usefulness to this day, a recent application of seniority is presented in Bose‐Einstein condensates of atoms with spin.