Nonexistence of a Positron—Hydrogen-Atom Bound State

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^-.