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Physical Review A

Classical spinning magnetic dipole in classical electrodynamics with classical electromagnetic zero-point radiation

Timothy H. Boyer

Published May 1, 1984• 13 citations
Publisher

Abstract

A classical spinning magnetic dipole is considered within classical electrodynamics with classical electromagnetic zero-point radiation. The stationary probability distribution for the angle of alignment between the spinning magnetic dipole and an external magnetic field is calculated when the system is located in an arbitrary spectrum of random classical radiation. It is found that for the Rayleigh-Jeans spectrum the probability distribution for alignment is just the Boltzmann distribution. However, in classical zero-point radiation the alignment probability distribution is independent of the magnetic field causing alignment. Moreover, for large classical spin angular momentum $S>>\ensuremath{\hbar}$, the average component of the spin in the direction of alignment is given by $〈{S}_{z}〉=S\ensuremath{-}\frac{1}{2}\ensuremath{\hbar}$, where $\ensuremath{\hbar}$ is Planck's constant (divided by $2\ensuremath{\pi}$) used to set the scale of the classical zero-point radiation spectrum. The results seem suggestive of the idea of space quantization in quantum theory. The model discussed here was first considered by S. Sachidanandam (unpublished).

DOI
10.1103/physreva.29.2389

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A classical spinning magnetic dipole is considered within classical electrodynamics with classical electromagnetic zero-point radiation. The stationary probability distribution for the angle of alignment between the spinning magnetic dipole and an external magnetic field is calculated when the system is located in an arbitrary spectrum of random classical radiation. It is found that for the Rayleigh-Jeans spectrum the probability distribution for alignment is just the Boltzmann distribution. However, in classical zero-point radiation the alignment probability distribution is independent of the magnetic field causing alignment. Moreover, for large classical spin angular momentum S>>\ensuremathℏS>>\ensuremath{\hbar}S>>\ensuremathℏ, the average component of the spin in the direction of alignment is given by 〈Sz〉=S\ensuremath−12\ensuremathℏ〈{S}_{z}〉=S\ensuremath{-}\frac{1}{2}\ensuremath{\hbar}〈Sz​〉=S\ensuremath−21​\ensuremathℏ, where \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ is Planck's constant (divided by 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ) used to set the scale of the classical zero-point radiation spectrum. The results seem suggestive of the idea of space quantization in quantum theory. The model discussed here was first considered by S. Sachidanandam (unpublished).
Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the zzz axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.
An external magnetic field is applied to the classical hydrogen atom treated in classical electromagnetic theory including classical electromagnetic zero-point radiation. In an earlier article, it was shown that the average value of each of the electron's action variables appears as a discrete value, corresponding to a representation of the rotation group, because of resonance between the periodic orbit of the electron and the random classical zero-point radiation. Here it is shown that, in the presence of a magnetic field and because of resonance, the classical orbital motion of the electron is in resonance with random classical zero-point radiation only for orientations of the orbit which take integer values for the angle made with the direction of the magnetic field, but excluding the m=0 orientation where the magnetic field is parallel to the orbital plane of the electron. Classical electromagnetic explanations are given for the Stern-Gerlach result, and for the Zeeman effect.
Abstract This paper is a study of a statistical ensemble of classical harmonic oscillators which is stationary in time, and whose position and momentum distribution functions are those of the corresponding quantum-mechanical oscillator in its ground state. If the oscillating particle is charged, then in order to maintain the distribution stationary a certain random electromagnetic field must be present. The intensity distribution of the radiation field is calculated, and it is found to be identical with the ‘photon vacuum’ of quantum electrodynamics. It is suggested, therefore, that this radiation field, which in quantum field theory is treated entirely formally, might exist in the classical sense. The method is extended to the excited states. It is found that the classical ensembles corresponding to these have probability distributions which may be negative. However, when attention is shifted to the quantum-mechanical ‘mixture’, this is no longer the case. Furthermore, the intensity distribution of the radiation field at temperature T is shown to be simply the sum of the Planck distribution and the previously obtained zero-temperature field. The application of these results to statistical mechanics is discussed. In this paper the treatment is non-relativistic throughout, but radiative corrections of the type which give rise to the Lamb shift are an integral part of the theory.
The Wigner representation of angular-momentum orientation in the classical limit on $J\ensuremath{\gg}1$ has been used to derive an equation of angular-momentum motion in the case of a closed two-level system interacting with an arbitrary polarized radiation. This equation enables us to carry out a theoretical analysis of angular-momentum motion and consider final states of the quantum system for all types of dipole transitions. We have found that in the case of a zero magnetic field in the final state of the system with $\stackrel{\ensuremath{\rightarrow}}{J}J$ or $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ optical transition, the angular momentum is directed along or opposite to the wave vector of the radiation depending on the sign of the polarization ellipsity, while for a linear polarization the angular momentum is isotropically distributed on the plane orthogonal to the polarization vector. For the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition two directions of angular momentum are possible in the final state. These directions are determined by the ellipsity of polarization. The spatial size of the final distribution of the angular momentum has been defined by the quantum uncertainty, 1/J,1/\sqrt{J},1/J​, of the angular-momentum orientation. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ or $\stackrel{\ensuremath{\rightarrow}}{J}J,$ transition particles come into ``dark'' states while for the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition they occupy the ``brightest'' state. In the presence of a nonzero magnetic field, particles with $\stackrel{\ensuremath{\rightarrow}}{J}J$ transition have in the final state only one angular-momentum orientation, i.e., along or opposite to the direction of the magnetic field depending on the polarization ellipsity sign. For the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition both directions are possible, i.e., along and opposite to the magnetic field. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition the directions of the angular momentum form a cone around the magnetic field.
We extend the recently introduced purely vibrational, classical field model of the hydrogen atom in two essential directions. First, the electron scalar field is replaced by a classical Dirac spinor field, which naturally generates intrinsic spin angular momentum and, in the non-relativistic limit, yields the Pauli equation with the correct spin–orbit coupling and Darwin term; the resulting fine-structure corrections match the standard quantum mechanical formula. Second, we incorporate a classical, Lorentz-invariant zero-point electromagnetic radiation background and show that, when a measurement apparatus is modelled as a resonant detector, the time averaged detection probability for a given stationary vibrational mode is proportional to its energy fraction. The Born rule thus emerges as a statistical consequence of deterministic evolution with zero-point noise, without any fundamental quantum postulate. The combined model reproduces both the full hydrogen spectrum (including fine structure) and the probabilistic predictions of quantum mechanics from purely classical field equations.