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Journal of Physics A Mathematical and General

Coefficients of fractional parentage for the states of an arbitrary number of j=1 bosons

Z. Marić, M. Popović‐Božić

Published Sep 1, 1980• 1 citation
Publisher

Abstract

The construction of symmetric states of N particles with angular momentum j=1 is presented. Elliott's result (1958), dealing with the angular momentum content of the symmetric subset of states belonging to the set of states of N particles with angular momentum j=1 is obtained using Racah algebra for the angular momentum coupling. The explicit functional dependence, on the total angular momentum, of the coefficients of fractional parentage (CFP) associated with one, two, three and four removed particles is found for arbitrary values of N. This dependence is expressed through the simple algebraic functions of J and N.

DOI
10.1088/0305-4470/13/9/015

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•Jan 1, 2013•0 citations
The construction of symmetric states of N particles with angular momentum j=1 is presented. Elliott's result (1958), dealing with the angular momentum content of the symmetric subset of states belonging to the set of states of N particles with angular momentum j=1 is obtained using Racah algebra for the angular momentum coupling. The explicit functional dependence, on the total angular momentum, of the coefficients of fractional parentage (CFP) associated with one, two, three and four removed particles is found for arbitrary values of N. This dependence is expressed through the simple algebraic functions of J and N.
Abstract In the independent particle model, the total wave function is expressed as a linear combination of products of single particle wave functions. Fractional parentage coefficients give the part of the expression depending on the symmetry properties of the wave function. An operator is introduced which picks out the antisymmetric part of any n-particle wave function. The operator is given explicit form, and an explicit formula for the fractional parentage coefficients is deduced. The total fractional parentage coefficients can be split into a part depending on the orbital vectors, and a part depending on the spin (and isotopic spin) vectors. The separate parts can have symmetries other than total antisymmetry. Operators are introduced which pick out states of definite symmetry from any state. The operators are given explicit form, and explicit formulae for orbital, or spin, fractional coefficients are deduced. Detailed results are given for the three, four and five particles cases.
Abstract A new procedure for large-scale calculations of the coefficients of fractional parentage (CFP) for many-particle systems is presented. The approach is based on a simple enumeration scheme for antisymmetric N particle states, and we suggest an efficient method for constructing the eigenvectors of two-particle transposition operator PN1,NP_{N_1 ,N}PN1​,N​ in a subspace where N 1 and N 2 = N − N 1 nucleons basis states are already antisymmetrized. The main result of this paper is that according to permutation operators PN1,NP_{N_1 ,N}PN1​,N​ eigenvalues we can distinguish totally asymmetrical N particle states from the other states with lower degree of asymmetry.
Simple and efficient methods for adding angular momenta and for finding angular momentum eigenstates for systems of equivalent electrons are developed. Seniority quantum numbers and coefficients of fractional parentage are presented in a simple manner which does not require 6- and 9-j symbols or knowledge of group theory. Several different common representations are used in specific examples. The transformation matrices between jj- and LS-coupled states are explicitly introduced as an example of change of representation and are used to find matrix elements of the spin-orbit interaction and to examine the Zeeman effect in the two representations. The material is suitable for a graduate course in quantum mechanics.
A new procedure for large-scale calculations of the coecients of fractional parentage (CFP) for manyparticle systems is presented. The approach is based on a simple enumeration scheme for antisymmetric N particle states, and we suggest an ecient method for constructing the eigenvectors of two-particle transposition operator PN1;N in a subspace where N1 and N2 = N N1 nucleons basis states are already antisymmetrized. The main result of this paper is that according to permutation operators PN1;N eigenvalues we can distinguish totally asymmetrical N particle states from the other states with lower degree of asymmetry.