Nonexistence of a Positron—Hydrogen-Atom Bound State

Abstract

One needs a necessary condition (rather than a sufficient condition, such as that of Rayleigh-Ritz) to even attempt to prove that a given bound state cannot exist. We use an adiabaticlike method to show that a positron (e+) and a hydrogen atom (H) cannot form a bound state. The separation r (but not r→) of the proton (p) and e+ is fixed; working in a subspace of zero total angular momentum, we calculate the lowest energy Ee1(r) of the electron in the field of e+ and p. An effective one-body p+e+ potential is then defined by V(1)(r)=Ee1(r)+e2r+e22a0. The necessary condition for the existence of a bound state of the true H+e+ system is the existence of a bound state of an artificial p+e+ system with an interaction V(1)(r). This one-body problem is readily solved, and we find that it (and therefore the true problem) has no bound state. The proof is not rigorous since Ee1(r) is not determined exactly, but the accuracy attained is such as to make the existence of an H+e+ bound state extremely unlikely. A by-product of the calculation is the determination of an improved lower bound on the ground-state energy of H−.