arXiv 2101.022022021-01-05quant-phhep-thnlin.PS

氢原子：电子自场的考虑

The hydrogen atom: consideration of the electron self-field

Leon V. Biguaa, Vladimir V. Kassandrov

备注：: 23 pages, 8 figures, 2 tables
发表信息：: Physics of Particles and Nuclei, 2020, V. 51, No.5, pp. 965-978
DOI：: 10.1134/S1063779620050020

摘要：

本文证明了在相对论量子力学中氢的正则问题中需要考虑电子的本征电磁场。从数学观点看，这一目标等价于确定Dirac方程组和麦克斯韦方程组（具有外库仑势）的自洽系统的处处正则解的谱.我们证明，只有特定的类的解决方案，“非线性”类似物的s-和p-状态，可以通过分解的一系列的解决方案，相对于精细结构常数参数$α$。在$α\rightarrow 0$的零近似下，薛定谔-泊松方程的自洽非相对论系统发生了还原。对于后者，我们用数值方法和变分方法得到了相应于基态和激发态的解。光谱的结合能与显着的精度再现的“Bohrian”依赖$W_n = W/ n^2$。因此，电离能$W$被证明是普适的，但比它的观测值小大约两倍。考虑了重正化过程的可能性和对$α^2$级结合能的相对论修正的考虑问题

英文摘要：

We substantiate the need for account of the proper electromagnetic field of the electron in the canonical problem of hydrogen in relativistic quantum mechanics. From mathematical viewpoint, the goal is equivalent to determination of the spectrum of everywhere regular solutions to the self-consistent system of Dirac and Maxwell equations (with external Coulomb potential). We demonstrate that only particular classes of solutions, "nonlinear" analogues of s- and p-states, can be obtained through decomposition of a solution in a series, with respect to the fine structure constant parameter $α$. In the zero approximation at $α\rightarrow 0$ the reduction to the self-consistent non-relativistic system of Schrödinger-Poisson equations takes place. For the latter, using both numerical and variational methods, we obtain the solutions corresponding to the ground and set of excited states. Spectrum of the binding energies with remarkable precision reproduces the "Bohrian" dependence $W_n = W/ n^2$. For this, the ionization energy $W$ proves to be universal yet about two times smaller than its observed value. Possibility of the renormalization procedure and the problem of account for relativistic corrections to the binding energies of order $α^2$ are considered

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