Historical research guide · 0886 · 2023
Two New Methods in Stochastic Electrodynamics for Analyzing the Simple Harmonic Oscillator and Possible Extension to Hydrogen
Daniel C. Cole
In one minute
Daniel Cole of Boston University works in stochastic electrodynamics, a theory that keeps classical physics intact and adds one ingredient: real electromagnetic noise filling space that does not switch off at absolute zero. Put a charged particle on a spring inside that noise, let it obey Maxwell’s equations and the classical law of motion including radiation reaction, and ask where the particle is likely to be found. Cole gives two new ways to get the answer from a single master formula that averages over every random field coefficient. The first route integrates them all out by brute force; the second notices that the master formula obeys a partial differential equation, which is far quicker to solve. Both give the same bell curve, and its width is exactly the value quantum mechanics gives for a harmonic oscillator at temperature T, with no quantum assumption used anywhere. He then sets out the harder target — doing the same for a classical hydrogen atom, which remains an open problem.
Connection
Chapter 2 says the vacuum is a real structured medium rather than an empty stage; stochastic electrodynamics is the cleanest test of that claim, because it derives quantum-looking answers from a classical particle in a classical zero-point field. Chapter 3 needs the next step Cole names — the classical hydrogen atom held in balance with that field, which is the mechanism behind the whole zero-point account of matter, inertia and gravity.
From the source
In stochastic electrodynamics the vacuum is modelled as a real classical electromagnetic field that is still there at zero temperature; its zero-point spectrum, in which each mode carries an energy proportional to Planck’s reduced constant times its frequency, is fixed by requiring the spectrum to be Lorentz invariant, as Marshall and Boyer showed, and independently by imposing the thermodynamic definition of absolute zero, as Cole showed.Section 1 and Section 2.1, around equation 14
For a charged particle bound in a harmonic well and driven by that field, the fully classical calculation gives a Gaussian position distribution whose mean square displacement equals Planck’s reduced constant divided by twice the mass times the natural frequency, multiplied by the hyperbolic cotangent of the ratio of the quantum of energy to twice the thermal energy — and at zero temperature this reduces exactly to the quantum ground-state value. No quantum assumption enters the derivation.Section 2.1, equations 38 and 39
Both results follow from one master expression, equation 1, which integrates a Dirac delta function of the particle’s trajectory over the probability distribution of every Fourier coefficient of the radiation field; Cole evaluates it two ways, by carrying out all the integrations explicitly in Section 2.1 and by showing in Section 3 that the same expression satisfies a diffusion-type partial differential equation in which the variance of position plays the role usually played by time.Equation 1 and equation 47
The same master expression yields other quantities of the oscillator: the joint probability density for position and momentum, the probability density for the oscillator’s kinetic plus potential energy, and the two-time position correlation, which in the continuum and resonant approximation equals the mean square displacement multiplied by the cosine of the natural frequency times the time separation.Section 2.2, equations 41 to 46
The real target is hydrogen, and Cole states why it is hard: every atomic system in nature is bound by the Coulomb force, which makes the classical electron’s equation of motion nonlinear, so no analytic trajectory of the kind used for the oscillator exists. The master expression should still be correct for classical hydrogen, but new analytic or numerical methods are needed to evaluate it, and no researcher has yet found one.Section 4, Concluding Remarks
Three independent simulation efforts — Cole and Zou in 2003, and Nieuwenhuizen and Liska more recently — all find that the classical electron in a zero-point field does not spiral into the nucleus, so stochastic electrodynamics resolves the old atomic-collapse problem of Rutherford’s classical atom, although a fraction of each ensemble ionises and the system’s chaotic dynamics amplifies small numerical errors.Section 4, Concluding Remarks
The way in
https://doi.org/10.3390/physics5010018How to cite it
Daniel C. Cole (2023) Two New Methods in Stochastic Electrodynamics for Analyzing the Simple Harmonic Oscillator and Possible Extension to Hydrogen doi:10.3390/physics5010018