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arXiv:2610.22393

Advanced computational methods for simulating the classical hydrogen atom in stochastic electrodynamics

Daniel C. Cole

Published Oct 13, 2026• 0 citations
arXivHTMLPDF

Abstract

Stochastic electrodynamics (SED) is a theory of nature that seeks to derive key aspects of quantum mechanics and quantum electrodynamics from a purely classical physics perspective. While SED has been successful in many areas, it has largely failed to stably simulate the 3D classical hydrogen atom. The computational methodology presented here overcomes these foundational challenges. We develop a pseudo-wave formulation that integrates the solid angle in k-space utilizing symmetry, resulting in a significantly reduced memory footprint and lower computational overhead for generating the zero-point radiation spectrum. This optimization is paired with a logarithmic frequency grid, a sliding window tracking the instantaneous Keplerian frequency, and a Tukey smoothing function to eliminate discrete force discontinuities. Integrated via an adaptive RKF45 solver across highly dense grids of up to 100-million bins, these techniques demonstrate the asymptotic convergence of ensemble stability. We show that increasing the grid resolution in the high-frequency domain systematically mitigates artificial electron decay, yielding a simulated probability density function that closely aligns with the quantum mechanical ground state P(r).

DOI
10.48550/arxiv.2610.22393

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Stochastic electrodynamics (SED) is a classical theory of nature advanced significantly in the 1960s by Trevor Marshall and Timothy Boyer. Since then, SED has continued to be investigated by a very small group of physicists. Early investigations seemed promising, as SED was shown to agree with quantum mechanics (QM) and quantum electrodynamics (QED) for a few linear systems. In particular, agreement was found for the simple harmonic electric dipole oscillator, physical systems composed of such oscillators and interacting electromagnetically, and free electromagnetic fields with boundary conditions imposed such as would enter into Casimir‐type force calculations. These results were found to hold for both zero‐point and non‐zero temperature conditions. However, by the late 1970s and then into the early 1980s, researchers found that when investigating nonlinear systems, SED did not appear to provide agreement with the predictions of QM and QED. A proposed reason for this disagreement was advocated by Boyer and Cole that such nonlinear systems are not sufficiently realistic for describing atomic and molecular physical systems, which should be fundamentally based on the Coulombic binding potential. Analytic attempts on these systems have proven to be most difficult. Consequently, in recent years more attention has been placed on numerically simulating the interaction of a classical electron in a Coulombic binding potential, with classical electromagnetic radiation acting on the classical electron. Good agreement was found for this numerical simulation work as compared with predictions from QM. Here this worked is reviewed and possible directions are discussed. Recent simulation work involving subharmonic resonances for the classical hydrogen atom is also discussed; some of the properties of these subharmonic resonances seem quite interesting and unusual.

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Stochastic electrodynamics (SED) is a theory of nature that seeks to derive key aspects of quantum mechanics and quantum electrodynamics from a purely classical physics perspective. While SED has been successful in many areas, it has largely failed to stably simulate the 3D classical hydrogen atom. The computational methodology presented here overcomes these foundational challenges. We develop a pseudo-wave formulation that integrates the solid angle in k-space utilizing symmetry, resulting in a significantly reduced memory footprint and lower computational overhead for generating the zero-point radiation spectrum. This optimization is paired with a logarithmic frequency grid, a sliding window tracking the instantaneous Keplerian frequency, and a Tukey smoothing function to eliminate discrete force discontinuities. Integrated via an adaptive RKF45 solver across highly dense grids of up to 100-million bins, these techniques demonstrate the asymptotic convergence of ensemble stability. We show that increasing the grid resolution in the high-frequency domain systematically mitigates artificial electron decay, yielding a simulated probability density function that closely aligns with the quantum mechanical ground state P(r).
Stochastic electrodynamics (SED) is a classical theory of nature advanced significantly in the 1960s by Trevor Marshall and Timothy Boyer. Since then, SED has continued to be investigated by a very small group of physicists. Early investigations seemed promising, as SED was shown to agree with quantum mechanics (QM) and quantum electrodynamics (QED) for a few linear systems. In particular, agreement was found for the simple harmonic electric dipole oscillator, physical systems composed of such oscillators and interacting electromagnetically, and free electromagnetic fields with boundary conditions imposed such as would enter into Casimir‐type force calculations. These results were found to hold for both zero‐point and non‐zero temperature conditions. However, by the late 1970s and then into the early 1980s, researchers found that when investigating nonlinear systems, SED did not appear to provide agreement with the predictions of QM and QED. A proposed reason for this disagreement was advocated by Boyer and Cole that such nonlinear systems are not sufficiently realistic for describing atomic and molecular physical systems, which should be fundamentally based on the Coulombic binding potential. Analytic attempts on these systems have proven to be most difficult. Consequently, in recent years more attention has been placed on numerically simulating the interaction of a classical electron in a Coulombic binding potential, with classical electromagnetic radiation acting on the classical electron. Good agreement was found for this numerical simulation work as compared with predictions from QM. Here this worked is reviewed and possible directions are discussed. Recent simulation work involving subharmonic resonances for the classical hydrogen atom is also discussed; some of the properties of these subharmonic resonances seem quite interesting and unusual.
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.
The article “Insights from the classical atom,” by Petar Grujić and Nenad Simonović (Physics Today, May 2012, page 41), was interesting but incomplete. It left out stochastic electrodynamics (SED), an approach to classical mechanics that includes the effects of a proposed zero-point radiation background.The zero-point radiation is a Lorentz-invariant random distribution of energy into different modes of electromagnetic radiation. Quantum electrodynamics (QED) uses the hypothesis that the amplitudes of the Lorentz-invariant radiation are quantized, whereas SED uses the hypothesis that they are continuous.The motions of electric charges in SED are modeled using the equations of motion for a particle, often associated with a concrete visualization. SED is considered part of classical physics because of the continuous distribution of amplitudes. However, discrete values for energy and momentum emerge from SED without the ad hoc quantization conditions of QED.Stochastic electrodynamics has been investigated as a classical physics explanation for quantum phenomena. For instance, the blackbody radiation spectrum has been derived without quantum assumptions.11. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374 Lorentz-invariant radiation provides perturbations that mimic quantum phenomena such as the Casimir force, the van der Waals force, the Lamb shift, and the radius of the Bohr atom.22. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737Physical chemistry provides many mathematical expressions that are often thought to be fundamentally quantum in nature. However, some of those expressions have been shown to emerge from both QED and SED. For example, the Schrödinger equation has been derived from SED,33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 and the wavefunction in the SED Schrödinger equation is shown to have the a priori significance of position probability amplitude. A physical model based on SED has been developed to increase our understanding of both the diffraction of particle beams and the Pauli exclusion principle.44. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707Quantum mechanics is often cited to justify chemical bonding on an elementary level. However, the visualizations used for chemical bonding are often based on mathematical expressions that can be derived from either QED or SED. Regardless of which turns out to be more accurate in physics experiments or more complete in mathematical description, the two theories overlap with regard to physical chemistry. Therefore, I propose that introductory chemistry taught in secondary schools can be rationalized as easily by references to SED as to QED.Either theory produces identical predictions for many experiments involving second-order correlations, squeezing (quadrature noise reduction), and the original Einstein-Podolsky-Rosen proposal.55. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912 However, SED is not completely equivalent to QED on a mathematical or physical level. The SED Schrödinger equation is incomplete, because there is a companion equation that has no counterpart in ordinary quantum mechanics and restricts initial conditions.33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 Furthermore, the predictions of QED and SED are different with regard to high-order correlations in nonlinear optics.I agree fully with Grujić and Simonović that the classical atom is alive and well. Models of the atom based on SED are sufficiently accurate for many technological and most pedagogical applications. Even if the classical atom isn’t the most fundamental description of the physical world, it is still an important one.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374, Google ScholarCrossref, ISI2. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737, Google ScholarCrossref, ISI3. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 , Google ScholarCrossref, ISIG. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993, , Google ScholarCrossref, ISI4. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707, Google ScholarCrossref, ISI5. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912, Google ScholarCrossref, ISI© 2013 American Institute of Physics.
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.
This paper develops and verifies a physics-first description of the hydrogen-atom transition from the L shell (n = 2) to the K shell (n = 1) by combining the historical Bohr model of atomic structure with classical Maxwell electrodynamics. Beginning with the Coulomb force, centripetal dynamics, and Bohr’s angular-momentum quantization condition, the analysis derives the allowed orbital radii, electron velocities, angular momenta, kinetic energies, electrostatic potential energies, and total bound-state energies of the hydrogen atom. For the n = 2 → n = 1 transition, the calculated energy difference is approximately 10.2 eV, corresponding through the Planck–Einstein relation ΔE = hν to a frequency of approximately 2.47 × 10¹⁵ Hz and a wavelength near 121.6 nm, consistent with the hydrogen Lyman-α transition. The emitted radiation is then examined through Maxwell’s equations and the vacuum constitutive relations. The resulting electromagnetic-wave description yields the transverse electric and magnetic fields together with the associated electromagnetic energy density, Poynting energy flux, momentum density, radiation pressure, and Maxwell stress tensor. In this way, the atomic energy difference obtained from the Bohr model is connected quantitatively to the classical propagation properties of the outgoing electromagnetic radiation. The derivation is mathematically consistent as a historical Bohr–Maxwell reconstruction, but it does not constitute a closed dynamical theory of spontaneous emission. The essential bridge between discrete atomic energy levels and the continuous Maxwell field is supplied by the quantum frequency condition ΔE = hν. Neither classical Maxwell electrodynamics nor Newtonian orbital mechanics independently derives this relation, and classical Maxwell theory does not by itself quantize the emitted radiation energy into individual photons. A complete treatment of spontaneous emission, transition probabilities, radiative lifetimes, and photon-field quantization requires quantum mechanics and ultimately quantum electrodynamics. Accordingly, the framework is presented as a verified correspondence between established limiting descriptions rather than as a replacement for Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics. By clearly separating the atomic bound-state calculation, the quantum energy-frequency bridge, and the classical electromagnetic propagation problem, the analysis provides a rigorous conceptual and mathematical foundation for subsequent Maxwell–Schrödinger, Dirac–Maxwell, quantum-electrodynamic, Maxwell–Scretching, and Scretching Quantum Chain formulations of hydrogenic radiative transitions.