Historical research guide · 0491 · 2015

Discrete Excitation Spectrum of a Classical Harmonic Oscillator in Zero-Point Radiation

Wayne Cheng-Wei Huang · Herman Batelaan

In one minute

Quantum mechanics is usually taught with a rule attached: energy comes in steps, and no classical theory can produce them. Wayne Cheng-Wei Huang and Herman Batelaan, at the University of Nebraska-Lincoln, put that rule to a numerical test and found it too strong. They took an ordinary classical charged oscillator — a charge on a spring obeying Newton and Maxwell and nothing else — and immersed it in the classical zero-point field, the real random electromagnetic background that stochastic electrodynamics restores to classical physics, with Planck’s constant entering only as the factor that sets the field’s strength. Then they struck it with a single light pulse, swept the pulse frequency, and computed the energy absorbed. A bare classical oscillator answers at one frequency only. The same oscillator sitting in the zero-point field answers at the harmonics as well, in a discrete multi-peaked spectrum whose peak shapes and heights match the quantum calculation to within the numerical convergence error, on a background raised by half a quantum. The steps came from the vacuum.

Connection

Chapter 2 holds that the vacuum is a real structured field rather than an empty stage, and chapter 3 that the quantum behaviour of matter is that field at work. This is the sharpest form of the argument the library can point to: the discreteness usually taken as the signature of quantum mechanics is reproduced by a classical particle whose only added ingredient is the zero-point field. Read it with the two companion sheets from the same group, on the Gaussian ground state and on the coherence test, which mark the other two edges of the same record.

From the source

Upon excitation by a single pulse, a classical harmonic oscillator immersed in classical electromagnetic zero-point radiation, as described by random electrodynamics, exhibits a quantized excitation spectrum in agreement with that of the quantum harmonic oscillator — a result the authors note is interesting in view of the generally accepted idea that classical theories do not support quantized energy spectra.Abstract; Results, Figure 2
With the classical zero-point field switched off, the classical oscillator supports only a single resonance at its natural frequency; with the field present as a constant background perturbation it shows a discrete multi-resonance excitation spectrum at the harmonics, and its background energy is shifted up by half a quantum of the natural frequency.Results paragraph; Figure 2, comparing the quantum, classical-with-field and bare-classical panels
The agreement between classical zero-point-field theory and quantum mechanics is within the error of the numerical convergence, in the shape as well as the magnitude of the resonance peaks, with the quantum side obtained by solving Schrödinger’s equation with a high-order multipole expansion over twenty energy levels.Conclusion; method paragraph on the twenty-level matrix
The match survives a change of geometry: as the excitation pulse angle is turned from zero to a right angle relative to the oscillator’s motion, the energy of the classical oscillator in the zero-point field scales in the same way as that of the quantum oscillator, with no higher harmonic when the pulse is perpendicular and no excitation at all when it is parallel.Figure 3, pulse angles of zero, one sixth, one quarter, one third and one half of pi
The nonlinearity that produces the higher harmonics is supplied by the classical zero-point field together with the spatial dependence of the excitation pulse, which is why the dipole approximation is not sufficient and the spatial dependence of the pulse field has to be expanded to twentieth order — a different origin from the nonlinearity of the atomic one-over-r potential in Cole and Zou’s subharmonic work.Method, Equations 9 to 12; closing discussion
The authors name the next steps themselves: extending the numerical approach to atomic systems, and using it to test critically the recent claims that superposition and entanglement are supported by classical zero-point-field theories.Closing discussion, final paragraph

The way in

https://doi.org/10.1007/s10701-015-9866-9

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

Licence checked directly. The arXiv page for arXiv:1206.6891 carries the arXiv perpetual non-exclusive distribution licence, which is not a Creative Commons licence, and the only licence Springer registers for the published article is a text-and-data-mining licence, so this page carries the summary, the claims and the authors’ own abstract and sends the reader to the source. Published as Foundations of Physics 45, 333–353 (2015); the abstract below is the authors’ abstract from their arXiv preprint of the same work, which carries the earlier title ‘Quantized Excitation Spectrum of the Classical Harmonic Oscillator in Zero-Point Radiation’, because the publisher withholds the published abstract from indexing services. The summary, the claims and the locators below were written from that preprint text; the journal version is expanded to twenty-one pages.

How to cite it

Wayne Cheng-Wei Huang, Herman Batelaan (2015) Discrete Excitation Spectrum of a Classical Harmonic Oscillator in Zero-Point Radiation doi:10.1007/s10701-015-9866-9

Where it sits in the curriculum

What the vacuum is