arXiv 0803.22892008-03hep-th

相对论薛定谔理论中的非相对论正电子素谱

Non-Relativistic Positronium Spectrum in Relativistic Schroedinger Theory

M. Mattes, M. Sorg

备注：: 67 pages and 3 figures

摘要：

在相对论薛定谔理论的框架下，在非相对论近似下研究了正电子偶素的最低能级。由于在非相对论极限下很难找到量子场方程的精确解，本文基于类氢波函数（即多项式乘指数）建立了一个近似方案。对于任何近似阶数$\NN (\NN=0,1,2,3,...)$，都会产生一个近似的具有相关能量的正电子偶素解谱，这与标准量子理论（附录）中对正电子偶素的传统处理非常相似。对于最低近似阶的$(\NN=0)$，对\emph{groundstate}能量的预测与标准理论的传统预测完全一致。然而，对于更高的近似阶数$(\NN=1,2,3)$，相应的预测结果与常规结果相差（大致）$0,9 [eV]$，这证实了先前的误差估计是由于使用球对称近似而引起的。激发态需要应用高阶近似$(\NN>>3)$，因此不能用当前的阶数$(\NN\le 3)$充分描述。

英文摘要：

The lowest energy levels of positronium are studied in the non-relativistic 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 $\NN (\NN=0,1,2,3,...)$ 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 $(\NN=0)$ the RST prediction for the \emph{groundstate} energy exactly agrees with the conventional prediction of the standard theory. However for the higher approximation orders $(\NN=1,2,3)$, 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 $(\NN>>3)$ and are therefore not adequately described by the present orders $(\NN\le 3)$.

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