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Dimorphism of Ortho-Positronium in Relativistic Schrödinger Theory

The non-relativistic energy levels of ortho-positronium are calculated in the quadrupole and octupole approximations for the interaction potential. For this purpose, the RST eigenvalue problem of angular momentum is illustratively solved for the quantum numbers $\jO=0,1,2,3,4$ and $\bjz=\pm 1$. This eigenvalue problem admits \emph{ambiguous} solutions for $0<|\bjz|<\jO$ whereas the solutions for $\bjz=0$ and $\bjz=\pm \jO$ are \emph{unique}. In order to attain some (at least approximative) solutions of the energy eigenvalue problem one tries a factorized ansatz for the wave function and thus splits off the angular problem (with its ambiguous solutions) from the residual radial problem. The latter does, as usual, finally fix the energy eigenvalues. But it is just by this procedure that the ambiguity of the angular problem is transferred to most of the energy levels which thereby become doubled. The corresponding doubling energy amounts to (roughly) one percent of the total binding energy and is, however, of purely \emph{electric} origin, since magnetism is completely neglected. Indeed, the charge distributions of both positronium constituents (i.e.\ electron and positron) do inherit their ambiguity from the ambiguous solution of the angular eigenvalue problem ($\leadsto$ charge "dimorphism"); and naturally the dimorphic configurations must then possess slightly different interaction energies of the electrostatic type.