Abstract

We substantiate the need for a “nonperturbative” account of the self-interaction of the electron with its own electromagnetic field in the canonical hydrogen problem in relativistic quantum mechanics. Mathematically, the problem is reduced to determination of the spectrum of everywhere regular axially symmetric solutions to the self-consistent system of Dirac and Maxwell equations, the classical analog of operator equations of quantum electrodynamics, in the presence of an external Coulomb potential. We demonstrate that only particular classes of solutions, “nonlinear” analogs of s- and p-states, can be obtained via expansion of a solution in a series over the fine structure constant \(\alpha \). In the zero approximation for \(\alpha \to 0\), we have the reduction to the self-consistent nonrelativistic system of Schrödinger–Poisson equations. The solutions corresponding to the ground state and a large set of excited states are obtained for this system using both numerical and variational methods. The spectrum of binding energies with remarkable accuracy reproduces the “Bohrian” dependence \({{W}_{n}} = {W \mathord{\left/ {\vphantom {W {{{n}^{2}}}}} \right. \kern-0em} {{{n}^{2}}}}\). In this case, the ionization energy \(W\) proves to be universal, yet about twice as small as its observed value. The problem of calculation of relativistic corrections to the binding energies and a relation between the model and the ideas and methods of quantum electrodynamics are discussed.

中文翻译：

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氢原子：电子自场的考虑

摘要

我们证实了相对论量子力学中正则氢问题中电子与自身电磁场的自相互作用的“非摄动”解释。从数学上讲，该问题简化为确定存在外部库仑势的Dirac和Maxwell方程（量子电动力学算子方程的经典模拟）的自洽系统的所有规则轴对称解的频谱。我们证明，只能通过在精细结构常数\（\ alpha \）上展开一系列解来获得特定类的解决方案，即s和p状态的“非线性”类似物。在\（\ alpha \ to 0 \）的零近似中，我们简化了Schrödinger-Poisson方程的自洽非相对论系统。使用数值和变分方法为该系统获得对应于基态和大量激发态的解。结合能的光谱具有非凡的精确度，再现了“玻色子”依赖性\（{{W} _ {n}} = {W \ mathord {\ left / {\ vphantom {W {{{{n} ^ {2}}} }} \ right。\ kern-0em} {{{n} ^ {2}}}} \）。在这种情况下，电离能\（W \）被证明是通用的，但大约是其观测值的两倍。讨论了对结合能的相对论修正的计算问题，以及模型与量子电动力学的思想和方法之间的关系。