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Physical Review A

General properties of ground-state energies of relativistic two-body Hamiltonians with Coulomb and Yukawa potentials

É. Papp

Published Jan 1, 1986• 6 citations
Publisher

Abstract

Estimates and critical stability thresholds characterizing the ground-state energies have been established with the help of a quasi- classical generalization of the virial theorem proposed previously. This enables us to evaluate ground-state energies in terms of the minima of corresponding Hamiltonian dispersions. Relativistic (${m}_{1}$=${m}_{2}$=${m}_{0}$) and nonrelativistic two-body Hamiltonians with Coulomb and Yukawa potentials have been considered. For relativistic spin-(1/2) particles, as well as for the nonrelativistic ones, the calculations proceed by fixing to unity the underlying phase-space quantum.

DOI
10.1103/physreva.33.719

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Estimates and critical stability thresholds characterizing the ground-state energies have been established with the help of a quasi- classical generalization of the virial theorem proposed previously. This enables us to evaluate ground-state energies in terms of the minima of corresponding Hamiltonian dispersions. Relativistic (m1{m}_{1}m1​=m2{m}_{2}m2​=m0{m}_{0}m0​) and nonrelativistic two-body Hamiltonians with Coulomb and Yukawa potentials have been considered. For relativistic spin-(1/2) particles, as well as for the nonrelativistic ones, the calculations proceed by fixing to unity the underlying phase-space quantum.
We study the ground state energy and the critical screening parameter of the Yukawa potential in nonrelativistic quantum mechanics. After a short review of the existing literature on these quantities, we apply fifth-order perturbation theory to the calculation of the ground state energy, using the exact solutions of the Coulomb potential together with a cutoff on the principal number summations. We also perform a variational calculation of the ground state energy using a Coulomb-like radial wave function and the exact solution of the corresponding minimization condition. For not too large values of the screening parameter, close agreement is found between the perturbative and variational results. For the critical screening parameter, we devise a novel method that permits us to determine it to 10 digits. This is the most precise calculation of this quantity to date, and allows us to resolve some discrepancies between previous results.
The asymptotic properties of a Schrödinger wave function which represents the bound ground state of a system of three interacting particles are examined. It is assumed that the interaction can be described by a static potential which is the sum of three two-body potentials and one three-body potential, where the potentials have the property that if any one of the particles is separated from the other two by a distance which tends to infinity, then the part of the potential energy which depends on the position of that particle tends to zero. The problem is treated nonrelativistically. Decreasing exponential bounds on the ground-state three-body wave function are established in configuration space. It is shown that these bounds depend only on the masses of the three particles, on the ground-state energy of the three-body system, and on the lower bounds on the spectra of the Hamiltonians for the three two-body systems arising when one of the three particles is removed and the remaining two interact through the corresponding two-body potential. It is furthermore shown that the three-body wave function in momentum space admits of analytic continuation into a tube-region in complex momentum space, which tube-region is the product of real momentum space and a convex bounded region in imaginary momentum space. The tube-region is explicitly determined. The implications of the results for the theory of the vertex function in quantum field theory are discussed. The relevancy of the results obtained in this paper to the variational computation of energy levels and wave functions of three-body systems is also briefly discussed. In the course of the derivation some generalizations are pointed out; in particular, the final results remain valid for a class of more general interactions, which cannot be described by static potentials. Likewise, the results remain valid for a class of spin-dependent interactions.
In this paper, we examine quantum systems with relativistic dynamics. We show that for a successful description of these systems, the application of Galilei invariant nonrelativistic Hamiltonian is necessary. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. Finally, we introduce and investigate the Schrödinger equation with relativistic dynamics for two-particle systems with harmonic oscillator and Coulomb potentials.
In this paper I develop a relativistic Hamiltonian and Schr\"odinger equation for three particles using a generalization of the assumptions that give rise to Todorov's relativistic quasipotential equation for two particles. This formalism is treated by a method that leads to a generalization of the separable Lagrange equilateral-triangle solution from gravitational nonrelativistic to relativistic nongravitational bound-state problems in quantum as well as classical mechanics. As an example I derive an analytic solution for the spectrum of a three-body system bound by scalar Coulomb forces.
Various aspects of critically bound quantal ground states close to the threshold in one, two, and three dimensions are examined in nonrelativistic quantum mechanics using attractive square-well and delta-function potentials. The mathematical and physical reasons are presented for the nonoccurrence of the bound state in the case of three dimensions and its occurrence in one and two dimensions for infinitesimally small strengths and ranges of the potential. Also analyzed is the condition for critical binding in relativistic quantum mechanics using the Klein–Gordon equation in one, two, and three dimensions. Corresponding features for the case of the Dirac equation in one and three dimensions are briefly discussed.
The question of stability of a given quantum system made up of charged particles is of fundamental interest in atomic, molecular, and nuclear physics. In this work, the stability for the negatively charged positronium (Ps)-like ions or the three-body system ( Z e + , e − , e − ) with Yukawa potentials is studied using correlated exponential wavefunctions based on the Ritz variational method. We obtained the critical screening parameter μ C as a function of the continuously varied nuclear charge Z , the critical nuclear charge Z C as a function of the screening parameter μ , and the ionization energies in terms of the screening parameter μ and Z . The critical nuclear charge for the bare Coulomb system ( Z e + , e − , e − ) obtained using 700-term correlated exponential wavefunctions is in accord with the reported results. The ionization energy, μ C , and Z C for the Yukawa system ( Z e + , e − , e − ) exhibit interesting behaviors. The present study describes the possible nonexistence of Borromean binding as well as Efimov states. The possible existence of quasi-bound resonances states for the negatively charged screened Ps-like ions is briefly discussed.