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Physical Review A

Matrix elements of potentials forL=1 hyperspherical states

Nir Barnea, Victor B. Mandelzweig

Published May 1, 1990• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.41.5209

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•Jan 1, 1988•0 citations
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.
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.
Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these spaces. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.
We investigate the adiabatic eigenfunctions and eigenvalues (and therefore the effective potentials) associated with the hyperspherical harmonic expansion of the wave function for three particles on a line subject to \ensuremath{\delta}-function interactions. Because of the special separability of the potential matrix elements, we obtain the effective potentials as the solutions of a transcendental equation and we also obtain the eigenfunctions in closed form. We demonstrate that the long-range (large hyper-radius \ensuremath{\rho}) behavior of the effective potentials is qualitatively different from the long-range behavior of the potential matrix elements.
Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.