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Journal of Physics A Mathematical and General

Coefficients of fractional parentage for a double-l boson system

Hong-Zhou Sun, Mei Zhang, Qi-Zhi Han, Gui‐Lu Long

Published Jun 7, 1990• 6 citations
Publisher

Abstract

The generalised coefficients of fractional parentage (CFP) for a double-l boson system are defined. The formulae of CFP with seniority are presented when the states of this system are classified according to two group chains. The calculations are done in the scheme of second quantisation. The results show that the computation of CFP in this way is very efficient.

DOI
10.1088/0305-4470/23/11/019

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The generalised coefficients of fractional parentage (CFP) for a double-l boson system are defined. The formulae of CFP with seniority are presented when the states of this system are classified according to two group chains. The calculations are done in the scheme of second quantisation. The results show that the computation of CFP in this way is very efficient.
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.
Reordering of operators in the second quantization formalism is described by using state operators as basic operators, where the state operator is either a cluster of creation operators or that of annihilation operators specified by a set of quantum numbers of any representation. The commutation relation between state operators is simply expressed in terms of the coefficient of fractional parentage (cfp), the factor specific to state operator (de) composition. Wick’s theorem, prevalently applied to a string of creation and annihilation operators, is extended to a string of state operators. Similar extension is done for the contraction theorem.