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Constructing wave functions in a hyperspherical basis using parentage scheme of symmetrization

Lia Leon Margolin

Published Jan 1, 2020• 1 citation

Abstract

Investigating few-body systems with identical particles in a hyperspherical basis yields the problem of obtaining symmetrized hyperspherical functions from functions with arbitrary quantum numbers. This article solves the problem of hyperspherical basis symmetrization for four-, five-, and six- body systems using Parentage Scheme of Symmetrization. Parentage coefficients and transformation coefficients corresponding to the [4], [31], [22], [211], representations of S4 groups with specific quantum numbers are obtained, and relationship between parentage coefficients and transformation coefficients between different sets of Jacobi coordinates for six-body systems are derived. Young operators, acting on N = 4, 5, 6 body hyperspherical functions symmetrized with respect to (N-1) particles, are derived. The symmetrized N = 4, 5, 6 body hyperspherical functions are obtained with different values of quantum numbers. The connection between the transformation coefficients for the identical particle systems and the parentage coefficients is demonstrated and the corresponding formulas are introduced.

DOI: 10.1063/5.0008672

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Investigating few-body systems with identical particles in a hyperspherical basis yields the problem of obtaining symmetrized hyperspherical functions from functions with arbitrary quantum numbers. This article solves the problem of hyperspherical basis symmetrization for four-, five-, and six- body systems using Parentage Scheme of Symmetrization. Parentage coefficients and transformation coefficients corresponding to the [4], [31], [22], [211], representations of S4 groups with specific quantum numbers are obtained, and relationship between parentage coefficients and transformation coefficients between different sets of Jacobi coordinates for six-body systems are derived. Young operators, acting on N = 4, 5, 6 body hyperspherical functions symmetrized with respect to (N-1) particles, are derived. The symmetrized N = 4, 5, 6 body hyperspherical functions are obtained with different values of quantum numbers. The connection between the transformation coefficients for the identical particle systems and the parentage coefficients is demonstrated and the corresponding formulas are introduced.

Matrix elements of two-body operators between many-particle permutational symmetry-adapted functions in hyperspherical coordinates are constructed. The matrix elements are evaluated using the appropriate hyperspherical coefficients of fractional parentage, the Raynal-Revai coefficients, and the hyperspherical recoupling coefficients. We use the power expansion of the two-body operators and obtain an analytic expression for the matrix element of each term. These expressions are studied numerically. The results allow precise evaluations of two-body matrix elements for few-body calculations in nuclear, atomic, and molecular physics.

We develop a numerical technique to describe few-body systems. Correlated Gaussian basis functions are used to expand the channel functions in the hyperspherical representation. The method is proven to be robust and efficient compared to other numerical techniques. The method is applied to few-body systems with short-range interactions, including several examples for three- and four-body systems. Specifically, for the two-component, four-fermion system, we extract the coefficients that characterize its behavior at unitarity.

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.

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.