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

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.