Non-Relativistic Positronium Spectrum in Relativistic Schroedinger Theory

Abstract

The lowest energy levels of positronium are studied in the nonrelativistic approximation within the framework of Relativistic Schrödinger Theory (RST). Since it is very difficult to find the exact solutions of the RST field equations (even in the non-relativistic limit), an approximation scheme is set up on the basis of the hydrogen-like wave functions (i.e. polynomial times exponential). For any approximation order N (N = 0, 1, 2, 3, . . .) there arises a spectrum of approximate RST solutions with the associated energies, quite similarly to the conventional treatment of positronium in the standard quantum theory (Appendix). For the lowest approximation order (N = 0) the RST prediction for the groundstate energy exactly agrees with the conventional prediction of the standard theory. However for the higher approximation orders (N = 1, 2, 3), the corresponding RST prediction differs from the conventional result by (roughly) 0, 9 [eV ] which confirms the previous estimate of the error being due to the use of the spherically symmetric approximation. The excited states require the application of higher-order approximations (N >> 3) and are therefore not adequately described by the present orders (N ≤ 3).