Physical Review A

Matrix elements of two-body operators between many-body symmetrized hyperspherical states

Akiva Novoselsky, Nir Barnea

Published Apr 1, 1995• 17 citations

Abstract

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.

DOI: 10.1103/physreva.51.2777

Related papers

Matrix elements between vector hyperspherical states

Nir Barnea, Victor B. Mandelzweig

Three-body symmetrized vector hyperspherical harmonic functions, realizing irreducible representations of the permutation group of three particles, are constructed and expressed in terms of Wigner D functions. Matrix elements of arbitrary two-body potentials between these hyperspherical states, including their velocity-dependent parts, are calculated analytically and expressed through the sum of products of the Wigner 3-j symbols and explicitly written functions of the radial variable.

Physical Review ADec 1, 199112 citations

Matrix elements of potentials forL=1 hyperspherical states

Nir Barnea, Victor B. Mandelzweig

Three-body L=1 symmetrized hyperspherical harmonic functions, realizing irreducible representations of the permutation group of three particles, are expressed in terms of Wigner D functions. Matrix elements of arbitrary two-body potentials between these hyperspherical states, including their velocity-dependent parts, are calculated analytically and expressed through the sum of products of the Wigner 3-j symbols and explicitly written functions of the radial variable.

Physical Review AMay 1, 199029 citations

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

Lia Leon Margolin

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.

AIP conference proceedingsJan 1, 20201 citation

Reduced Matrix Elements of Tensor Operators

J S Briggs

Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.

Journal of Mathematical PhysicsApr 1, 19704 citations

Two-body-operator matrix elements in SU(n) for application to complex spectroscopy

R. D. Kent, M. Schlesinger

We present a derivation of closed-form algebraic expressions for two-body operators in the context of the special-unitary-group approach to complex spectroscopy. The method involves the use of a graphical representation of the permutational symmetry-adapted irreducible-representation basis which facilitates the decomposition of the matrix elements into simple factors. The results afford significant reductions of computational complexity in cases pertaining to higher-than-spin-1/2 N-particle systems.

Physical Review ANov 1, 19875 citations

ON THE CALCULATION OF ENERGY MATRIX FOR THE SYSTEM OF IDENTICAL PARTICLES WITH ORDERED SPIN

Itay Kaplan

S>Transformation permutation group matrices used in calculation of two- shell element configurations were developed and tabulated for possible symmetry groups with 3 to 6 total particle numbers. The table also includes the permutation matrix group in the expressions describing operator matrix elements for two-particle interactions G in the system found in the field with an arbitrary symmetry and non-conservation of motion moment. (R.V.J.)

Jan 1, 19630 citations