Problems with Translational Invariance of Three-Particle Systems

Abstract

A new method for calculation of non-relativistic energy spectrum of Coulomb three-body systems with two identical particles has been developed. The novelty of the method is the introduction of an expansion of the wave function on harmonic oscillator (HO) functions with different sizes in the Jacobi coordinates instead of only one unique size parameter in the traditional approach. The method presented obeys the principles of antisymmetry and translational invariance. The theoretical formulation has been illustrated by evaluation of ground state energies of a number of Coulomb three-body systems with two identical particles for zero HO excitation energy. The analytical solution of this problem in case of only one size parameter has been derived. The obtained results show significant advantage of the base with different sizes over the traditional approach for investigation of the bound state problem of quantum systems.

New possibilities of harmonic oscillator basis application for calculation of the ground state energy of a Coulomb non-identical three-particle system

“…Hence in these calculations nonantisymmetrized basis (8) can be used unlike the previously published method, which was devoted to the development of the HO expansion method for the Coulomb three-particle systems with two identical particles [13]. From the numerical point of view, the method for the threeparticle systems with two identical particles results in more complicated expressions for matrix elements of the intrinsic Hamiltonian as in the case of non-identical particles.…”

“…The matrix element of the first kinetic energy term of the Hamiltonian (7) between the coupled threeparticle HO functions (8) may be evaluated applying the usual Racah technique [21] (see details in [13]):…”

“…By contrast, the traditional approach actually provides only one unique oscillator length for all Jacobi coordinates. Recently, the HO basis with two intrinsic scales has been applied for evaluation of the ground state energy of a number of Coulomb three-body systems with two identical particles for zero HO excitation energy and shows a significant advantage over the traditional HO basis expansion [13]. It seems that intrinsic coordinates accompanied with individual variational size parameters are more adapted to better grasp the physical contents of the quantum systems.…”

Two-Parameter Harmonic Oscillator Expansion Method for Calculating Non-Relativistic Ground State Energy of Coulomb Three-Particle Systems with Two Identical Particles

“…In Ref. [2], a new variational method is proposed for calculating the energy spectrum of three-particle Coulomb systems, consisting of two identical particles and a different one. In this method, individual parameters for each Jacobi coordinate are introduced instead of only one such parameter in the traditional approach.…”

“…For a nonzero number of oscillator quanta it showed much higher accuracy and convergence rate than the calculation with one nonlinear variational parameter. Keeping in mind the advantages of introducing individual parameters for Jacobi coordinates in the case of systems of three different particles, in the present work we apply the method of [2] to the calculation of the ground state energy for a number of Coulomb systems, comprising two identical particles and a different one at nonzero numbers of oscillator quanta and report results of the comparison of its accuracy and convergence rate with those of the traditional calculation. Thanks to the availability of practically "exact" solutions for three-particle Coulomb e-mail: algirdas.deveikis@vdu.lt…”