Quantum Mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics
Daniel C. Cole · Yi Zou
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In one page
Daniel Cole and Yi Zou asked a question most physicists had stopped asking. Can ordinary classical physics build the hydrogen atom, if you add one ingredient — a real, random electromagnetic field filling all of space, the zero-point field? That programme is called stochastic electrodynamics, and by the early 1980s the verdict was that it failed, because the classical atom was expected to ionise. Cole and Zou reopened the case with a computer instead of an equation. They tracked one classical electron around a nucleus for long stretches of simulated time, with Maxwell’s equations, radiation reaction and roughly two million zero-point waves acting on it, then measured how long the electron spent at each distance from the nucleus. Eleven independent runs, about fifty-five days of processor time, no fitted parameters. The probability curve that emerges converges on the one Schrödinger’s equation gives for the ground state of hydrogen. Their own conclusion is careful: this does not prove stochastic electrodynamics, but the old impasse may have been a mathematics problem rather than a physics one.
Why it matters hereChapter 2 needs a vacuum that does real, measurable work on ordinary matter, and this is about the sharpest demonstration in the library — the shape of the hydrogen atom drawn out of a classical electron and the zero-point field alone, with nothing fitted. Chapter 13’s unified picture, in which quantum behaviour is the visible signature of a real background field rather than a separate set of rules, rests on results of exactly this kind.
What it claims
01A classical charged point particle, moving under a Coulomb binding potential and classical electromagnetic zero-point radiation alone, yields a radial probability density that agrees with the ground state of hydrogen given by Schrödinger’s wave equation — and it does so without any fitting parameters.Abstract; the four snapshots of Figure 2
Published and peer-reviewed02The stabilising mechanism is a frequency balance in the zero-point spectrum, as Boyer suggested in 1975: a wide orbit couples mainly to low-frequency, low-energy components and decays on average, while a tight orbit couples to high-frequency, higher-energy components and is driven outward, so the electron settles into a stochastic pattern around a preferred radius rather than spiralling in.Discussion following Figure 2, on Boyer 1975
Published and peer-reviewed03The zero-point field used is the Lorentz-invariant classical spectrum with energy density proportional to the cube of the frequency; its functional form follows either from requiring Lorentz invariance or from requiring that no heat flow during reversible thermodynamic operations.Opening section, on the derivation of the zero-point spectrum
Settled physics04The result comes from eleven independent simulations, each started at the Bohr radius of 0.53 ångström with a different random seed for the field, integrated with a fifth-order adaptive Runge-Kutta scheme against roughly 2.2 million plane waves, for a combined 55 days of processor time; the individual runs each resemble the quantum curve and the combined result matches it more closely.Simulation description and Figure 1
Published and peer-reviewed05Cole and Zou argue that the earlier consensus against stochastic electrodynamics — that a hydrogen atom would ionise at zero temperature and that its spectra would not match quantum mechanics — may have come from the difficulty of analysing nonlinear stochastic differential equations rather than from a physical flaw in the theory.Introduction, on the 1980s objections; and the paragraph beginning ‘For that reason’
What to watch06The authors state plainly what remains open: relativistic corrections and high-frequency effects, atomic spectra, many-electron systems, spin, and an account of how photon behaviour arises. Those are the measurements and calculations that would settle whether the zero-point field is the foundation of quantum mechanics.Closing two paragraphs
What to watch
The way in
https://doi.org/10.1016/j.physleta.2003.06.003Published as Physics Letters A volume 317, issues 1 to 2, pages 14 to 20, 6 October 2003, from the Department of Manufacturing Engineering at Boston University. LICENCE. The version of record is closed at the publisher and Unpaywall reports no open repository copy; the authors’ preprint carries the standard arXiv non-exclusive distribution licence, which is not a Creative Commons grant, so this page reproduces no text. SOURCE FOR THE CLAIMS. The full preprint, arXiv quant-ph/0307154 (22 July 2003), was fetched and read on 2026-09-08; every summary sentence, claim and locator below is written from it. AUTHORS. The registry record listed Daniel C. Cole alone; the paper is by two authors, Daniel C. Cole and Yi Zou, and both are named here. CHAPTERS. The skeleton carried chapter 2, chapter 3 and chapter 6; the paper is about the vacuum field setting an atomic state, so it is filed to chapter 2 and chapter 13.
How to cite it
Daniel C. Cole, Yi Zou (2003) Quantum Mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics. doi:10.1016/j.physleta.2003.06.003
Where it sits in the curriculum