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Physical Review

Direct Evaluation of Fractional Parentage Coefficients Using Young Operators: Normalization of Four-Particle Wave Functions

M. J. Englefield

Published Jun 1, 1955• 1 citation
Publisher

Abstract

In a recent paper by Jahn, four-particle wave functions of specified symmetry were generated by using Young operators. When three of the angular momenta of the individual particles were the same, the functions were not normalized, an undetermined factor ${N}_{\ensuremath{\alpha}\ensuremath{\beta}}$ being introduced. Explicit expressions for this factor, in terms of the Hope $\ensuremath{\chi}$ function, are given.

DOI
10.1103/physrev.98.1213

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In a recent paper by Jahn, four-particle wave functions of specified symmetry were generated by using Young operators. When three of the angular momenta of the individual particles were the same, the functions were not normalized, an undetermined factor N\ensuremathα\ensuremathβ{N}_{\ensuremath{\alpha}\ensuremath{\beta}}N\ensuremathα\ensuremathβ​ being introduced. Explicit expressions for this factor, in terms of the Hope \ensuremathχ\ensuremath{\chi}\ensuremathχ function, are given.
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.
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.
A procedure for the construction of nonspurious harmonic oscillator wave functions with arbitrary permutational symmetry has recently been proposed. The resulting wave functions are expressed in terms of normalized Jacobi coordinates, and involve a new type of harmonic oscillator coefficients of fractional parentage. A simple algorithm to transform these states from the Jacobi coordinates to the single particle coordinates is presented. This is a generalization to an arbitrary number of particles of the harmonic oscillator transformation from center-of-mass and relative coordinates to single particle coordinates.