The Hydrogen Atom: Consideration of the Electron Self-Field
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e Hydrogen Atom: Consideration of the Electron Self-Field L. V. Biguaaa, * and V. V. Kassandrovb, ** a
Quantum Technology Center, Moscow State University, Moscow, 119991 Russia Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, Moscow, 117198 Russia *e-mail: [email protected] **e-mail: [email protected]
b
Received March 2, 2020; revised March 13, 2020; accepted March 26, 2020
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 α . In the zero approximation for α → 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 Wn = W n2 . 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. Keywords: spinor electrodynamics, Dirac–Maxwell system of equations, Coulomb self-action, Schrödinger– Poisson system of equations, soliton-like solutions, Bohrian binding energy spectrum DOI: 10.1134/S1063779620050020
1. THE HYDROGEN ATOM AND CLASSICAL FIELD MODELS OF ELEMENTARY PARTICLES It is well known that the successful description of the hydrogen atom, first in Bohr’s theory, and then via solutions to the Schrödinger equation was one of the main motivations for the development of quantum theory. Later, Dirac described almost perfectly the observed hydrogen spectrum using his relativistic generalization of the Schrödinger equation. A slight deviation from experiment (the Lamb shift between 2s1/ 2 and 2 p1/ 2 levels in the hydrogen atom) discovered later was interpreted as the effect of electron interaction with vacuum fluctuations of electromagnetic field and explained in the framework of quantum electrodynamics (QED) using second quantization. Nonetheless, the problem of the description of the hydrogen atom still attracts attention of researchers, and, in some sense, is a “touchstone” for many new theoretical developments. In particular, it is possible to obtain the energy spectrum via purely algebraic methods [1]. Theories
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