Bound States of a Relativistic Two-Body Hamiltonian; Comparison with the Bethe-Salpeter Equation

Publisher Website
Google Scholar
Abstract
The tightly bound states of a simple relativistic two-body Hamiltonian are studied. The coupling constant necessary for obtaining a given binding energy is obtained numerically for a Yukawa-like interaction with variable range. Some fairly general relations restricting the connection between binding energy, coupling constant, and force range, expected to be valid in the tight-binding limit, are derived and tested. A comparison is made with results obtained by Schwartz for the "corresponding" Bethe-Salpeter (B-S) equation. It is concluded that, even in the strong-binding limit, the pair, multimeson, and retardation effects taken into account by the ladder-approximation B-S equation are not very important, at least as far as the relation between coupling constant and binding energy is concerned. These results suggest that a Hamiltonian of the type considered may be a useful tool in exploratory calculations involving quark models. In this connection, we show that a Yukawa-like interaction leads to relativistic motion in the tight-binding limit even if used in a Hamiltonian incorporating relativistic kinematics, and we thereby generalize a result of Greenberg's.