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Classical Zero-Point Radiation and Relativity: The Problem of Atomic Collapse Revisited

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Abstract

The physicists of the early twentieth century were unaware of two aspects which are vital to understanding some aspects of modern physics within classical theory. The two aspects are: (1) the presence of classical electromagnetic zero-point radiation, and (2) the importance of special relativity. In classes in modern physics today, the problem of atomic collapse is still mentioned in the historical context of the early twentieth century. However, the classical problem of atomic collapse is currently being treated in the presence of classical zero-point radiation where the problem has been transformed. The presence of classical zero-point radiation indeed keeps the electron from falling into the Coulomb potential center. However, the old collapse problem has been replaced by a new problem where the zero-point radiation may give too much energy to the electron so as to cause “self-ionization.” Special relativity may play a role in understanding this modern variation on the atomic collapse problem, just as relativity has proved crucial for a classical understanding of blackbody radiation.

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Notes

  1. Traditional classical electron theory is described by [6]. This volume is a republication of the second edition of 1915 based on Lorentz’s Columbia University Lectures of 1909. Note 6, p. 240, gives Lorentz’s explicit assumption on the boundary conditions.

  2. In the textbooks, classical electromagnetism is treated from the point of view of electromagnetic technology where the charge and current sources provide overwhelmingly large electromagnetic fields; classical electromagnetism is not treated as a subject relevant for atomic physics where the sources may give fields comparable to the background fields of thermal radiation or zero-point radiation. Accordingly, the homogeneous boundary conditions on Maxwell’s equations are rarely if ever mentioned in textbooks of classical electromagnetism.

  3. A review of the work on classical electromagnetic zero-point radiation up to 1996 is provided by [12].

  4. In 1975, I was well aware that use of a harmonic oscillator model gave exactly \(J=\hbar .\) However, this suggested an exact agreement with the quantum ground-state results for hydrogen. The harmonic-oscillator model result was published by [13]. However, neither the rotator model nor the oscillator model corresponds to particle orbits in a Coulomb potential, and so these models are at best suggestive qualitative approximations.

  5. See also [14]. These authors give a very nice spiral-down-time calculation for a classical electron in a circular orbit in hydrogen. They show that the “transition times” are in good agreement with those found from quantum theory.

  6. Nieuwenhuizen and Liska define ionization “as the moment when the electron stays above \(\mathcal E =-0.05\) for a duration for at least 10\(^{7}\)t\(_{0}\).” Here \(\mathcal E \) is given in Bohr units and t\(_{0}\) is the Bohr period.

  7. See Figs. 1 and 2 of Ref. [4].

  8. Here we have chosen action-angle variables differing by a factor of \(1/(2\pi )\) from those of problem 28 of Goldstein’s text and have written the strength of the potential as \(k=Ze^{2}.\)

  9. As another example of the interesting connections between classical and quantum theories, we note that the quantum mechanicalproblem of atomic collapse for large values of Z \(\gtrsim \) 137 is mirrored in the relativistic classical situation. When zero-point radiation is present, we find that \({\langle } J{\rangle } \approx {\hbar }.\) Thus, when Z exceeds 137, then \(Ze^{2}/({\hbar } c){\gtrsim }1,\) and the classical relativistic energy in Eq. (9) becomes complex, indicating trajectories spiraling into the Coulomb center. Apparently recent experimental work with graphene suggests confirmation of the ideas of quantum mechanical atomic collapse for large Z. See [22].

  10. See, for example, Eisberg and Resnick in Ref. [1], p. 12 or Krane in Ref [2], p. 80.

  11. D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan. The no-interaction theorem is referred to in Goldstein’s mechanics text, Ref. [21], pp. 332, 334, and 362.

  12. This adiabatic invariance corresponds to the familiar mechanical problem of slowly pulling a string through a hole in a frictionless table, with the string attached to a rotating block. The angular momentum J (an action variable) is conserved during the slow change of the strength of the central force.

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  1. Department of Physics, City College of the City University of New York, New York, 10031, USA

    Timothy H. Boyer

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Boyer, T.H. Classical Zero-Point Radiation and Relativity: The Problem of Atomic Collapse Revisited. Found Phys 46, 880–890 (2016). https://doi.org/10.1007/s10701-016-0008-9

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