Quadrupole Approximation for Para-Positronium in Relativistic Schr\"odinger Theory

The non-relativistic energy levels of para-positronium are calculated in the quadrupole approximation of the interaction potential. This approximation technique takes into account the anisotropy of the electrostatic electron-positron interaction in the lowest order. The states due to different values of the quantum number $(l_z)$ of angular momentum are found to be no longer degenerate as is the case in the conventional theory. The physical origin of this elimination of the conventional degeneracy may intuitively be attributed to the state-dependent inertial \emph{broadening} of the rotating charge clouds; the corresponding \emph{anisotropic} deformation (in the quadrupole approximation) lowers then the negative electrostatic interaction energy. The result of this influence of anisotropy is that the states with $l_z=0$ adopt smaller binding energy whereas the states with maximal value of $|l_z|$ (for fixed principal quantum number $n$) have the largest binding energy within the angular momentum multiplet $(-|l_{z,\mathrm{max}}| \le l_z \le |l_{z,\mathrm{max}}|)$. This yields a certain kind of electric fine-structure splitting with the splitted RST levels being placed in a relatively narrow band around the (highly degenerated) conventional levels.