Abnormal states with unequal constituent masses

Abstract

The Bethe–Salpeter equation for system of two oppositely charged particles not only reproduces the Coulomb spectrum, but, for enough large coupling constant C>π4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal C}>\\frac{\\pi }{4}$$\\end{document}, predicts additional levels not predicted by the Schrödinger equation. These relativistic states (called abnormal), in contrast to the normal ones, are dominated, for more than 90–99%, by Fock states involving the exchange particles—the photons, whereas contribution of two massive charged particles themselves is rather small (1–10%). Since the carrier of a large (positive) charge is a heavy ion, and the negative charge is provided by electron, the masses of two constituents are very different. It is shown that in a system with so different masses the abnormal states still exist. Moreover, the effect of unequal masses is attractive. The balance between photons and charged constituents is weakly sensitive to the mass ratio, so the photons still predominate.