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arXiv:1611.10200 · Entropy

On the Stability of Classical Orbits of the Hydrogen Ground State in Stochastic Electrodynamics

Theo M. Nieuwenhuizen

Published Apr 13, 2016• 11 citations
arXivHTMLPDF

Abstract

De la Peña 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics connect to a stable situation, where the electron neither collapses onto the nucleus nor gets expelled from the atom. Although the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Peña-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an 1 / r 2 potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium.

DOI
10.3390/e18040135

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De la Peña 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics connect to a stable situation, where the electron neither collapses onto the nucleus nor gets expelled from the atom. Although the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Peña-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an 1 / r 2 potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium.
Stochastic electrodynamics is a classical theory which assumes that the physical vacuum consists of classical stochastic fields with average energy 1 2 ℏ ω in each mode, i.e., the zero-point Planck spectrum. While this classical theory explains many quantum phenomena related to harmonic oscillator problems, hard results on nonlinear systems are still lacking. In this work the hydrogen ground state is studied by numerically solving the Abraham–Lorentz equation in the dipole approximation. First the stochastic Gaussian field is represented by a sum over Gaussian frequency components, next the dynamics is solved numerically using OpenCL. The approach improves on work by Cole and Zou 2003 by treating the full 3 d problem and reaching longer simulation times. The results are compared with a conjecture for the ground state phase space density. Though short time results suggest a trend towards confirmation, in all attempted modellings the atom ionises at longer times.
In a recent paper [arXiv:1502.06856] the authors studied numerically the hydrogen ground state in stochastic electrodynamics (SED) within the the non-relativistic approximation. In quantum theory the leading non-relativistic corrections to the ground state energy dominate the Lamb shift related to the photon cloud that should cause the quantum-like behaviour of SED. The present work takes these corrections into account in the numerical modelling. It is found that they have little effect; the self-ionisation that occurs without them remains present. It is speculated that the point-charge approximation for the electron is the cause of the failure.
In this paper, the harmonic oscillator problem in Stochastic Electrodynamics is revisited. Using the exact shape of the Lorentz damping term prevents run-away effects. After introducing a cut-off in the stochastic power spectrum and regularizing the stochastic force, all relevant integrals are dominated by resonance effects only and results are derived that stem from those in the quantum ground state. For an orbit with specific position and momentum at an initial time, the average energy and the average rate of energy change are evaluated, which stem with each other. Resonance effects are highlighted along the way. An outlook on the hydrogen ground state problem is provided.
The hydrogen ground state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work we consider the regularization of the noise in the hydrogen ground state problem. Several renormalization schemes are considered. Some are well behaved, while in other ones, the high frequency renormalization induces pathologies at low frequencies. In no situation we find a way to escape from the previously signaled self-ionization.
The fact that an orbiting electron does not collapse into the nucleus of its parent atom has thus far been considered a major obstacle to a classical interpretation of the stable states of orbiting electrons. Quantum mechanics avoids the very problem by discussing the probability of finding it, a method that confessedly has been very fruitful in the exploring the behaviour of elementary particles. Nonetheless, why should necessarily those two approaches be regarded as each others' enemies? In this paper it is shown that the classical mechanistic approach is still capable of explaining the eternal, circular movement of an electron around a nucleus. It is possible if reformulating the laws of action involved, returning to the simple electrostatic model, based upon Coulomb's law (1785, 1771). Further, in this paper it is also discussed, how electromagnetic radiation due to the de-excitation of excited electrons, orbiting around a positive nucleus, can be explained classically, as a sudden peak in the otherwise zero electric field, due to the inwards spiralling movement connected to the de-excitation. The concept of a distinct particle, the 'photon' is thereby rejected.