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

TL;DR

This paper calculates non-relativistic energy levels of para-positronium using a quadrupole approximation that accounts for anisotropic electrostatic interactions, revealing angular momentum-dependent energy splitting and lifting of degeneracy.

Contribution

It introduces a quadrupole approximation in relativistic Schrödinger theory to analyze para-positronium, highlighting anisotropic effects on energy levels and degeneracy lifting.

Findings

States with different $l_z$ are no longer degenerate.

Anisotropic deformation lowers the electrostatic energy for certain states.

Energy levels exhibit fine-structure splitting around conventional levels.

Abstract

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.