电子与质子间静电Bopp-Landé-Escheras-Podolsky相互作用下氢原子的薛定谔光谱
On the Schrödinger spectrum of a hydrogen atom with electrostatic Bopp-Landé-Thomas-Podolsky interaction between electron and proton
摘要：
一个氢原子的薛定谔光谱，被模拟为一个由一个点电子和一个点质子组成的两体系统，与20世纪40年代由波普、兰德-托马斯和波多尔斯基提出的库仑定律的修正相互作用（BLTP）。BLTP理论假设存在一个自然界的电磁长度尺度-波普长度$\varkappa ^{-1}$ -，大意是静电对相互作用只有在距离比$\varkappa^{-1}$短得多的情况下才明显偏离库仑定律。严格的下限和上限构造的薛定谔能级的氢原子，$E_{\ell,n}(\varkappa)$，所有的$\ell\in\{0,1,2,...\}$和$n >\ell$。还对能级$E_{0,1}(\varkappa)$、$E_{0,2}(\varkappa)$和$E_{1,2}(\varkappa)$进行了数值计算，并绘制了与$\varkappa^{-1}$的关系曲线。结果发现，BLTP理论预测的非相对论修正的Lyman-$α$线的分裂，除了其著名的相对论精细结构分裂。在假设这种分裂在相对论计算中不会消失的情况下，有人认为，目前对莱曼- $α$线的精确测量表明，$\varkappa^{-1}$必须小于$\approx 10^{-18}\,\mathrm{m}$。有限的质子尺寸效应被发现不修改这一结论。因此，基本点电荷的静电场能量，尽管在BLTP电动力学中是有限的，但远大于电子的经验静止能量。如果，正如电子的所有“重整化理论”中所假设的那样，物理电子的经验静止质量是其净静止质量加上静电场能量的总和（$/c^2$），那么在BLTP电动力学中，电子必须被分配一个负的裸静止质量。
英文摘要：
The Schrödinger spectrum of a hydrogen atom, modelled as a two-body system consisting of a point electron and a point proton, interacting with a modification of Coulomb's law proposed, in the 1940s, by Bopp, Landé--Thomas, and Podolsky (BLTP). The BLTP theory hypothesizes the existence of an electromagnetic length scale of nature --- the Bopp length $\varkappa ^{-1}$ ---, to the effect that the electrostatic pair interaction deviates significantly from Coulomb's law only for distances much shorter than $\varkappa^{-1}$. Rigorous lower and upper bounds are constructed for the Schrödinger energy levels of the hydrogen atom, $E_{\ell,n}(\varkappa)$, for all $\ell\in\{0,1,2,...\}$ and $n >\ell$. The energy levels $E_{0,1}(\varkappa)$, $E_{0,2}(\varkappa)$, and $E_{1,2}(\varkappa)$ are also computed numerically and plotted versus $\varkappa^{-1}$. It is found that the BLTP theory predicts a non-relativistic correction to the splitting of the Lyman-$α$ line in addition to its well-known relativistic fine-structure splitting. Under the assumption, that this splitting doesn't go away in a relativistic calculation, it is argued that present-day precision measurements of the Lyman-$α$ line suggest that $\varkappa^{-1}$ must be smaller than $\approx 10^{-18}\,\mathrm{m}$. Finite proton size effects are found not to modify this conclusion. As a consequence, the electrostatic field energy of an elementary point charge, although finite in BLTP electrodynamics, is much larger than the empirical rest energy of an electron. If, as assumed in all `renormalized theories' of the electron, the empirical rest mass of a physical electron is the sum of its bare rest mass plus its electrostatic field energy ($/c^2$), then in BLTP electrodynamics the electron has to be assigned a negative bare rest mass.
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