The classical linear oscillator in classical electrodynamics with classical zero-point radiation

Abstract

In this article, we consider a classical charged particle in a one-dimensional classical linear oscillator potential interacting with classical electromagnetic zero-point radiation. We show that the oscillator has a stable ground state and unstable resonant excited states which are analogous to those of the quantum harmonic oscillator. We develop six themes. 1) Classical electromagnetic zero-point radiation is Lorentz invariant and requires that any mechanical system which comes to equilibrium in it is at least approximately Lorentz invariant. Very few potentials meet this approximately-Lorentz-invariant criterion. 2) Classical relativistic waves travel at finite speed c, which requires that the interaction of a point charge e located at r(t) be treated using the full electric field E[r(t), t], not simply in the dipole approximation as E[0, t]. 3) The mechanical linear oscillator has SO(2) symmetry associated with the time behavior of the angle variable phi(t) = omega-zero t + phi-zero, and leads to representations associated with integer indices for the action variable J. 4) In the large-mass approximation, the ground state is stable in classical zero-point radiation, even when higher radiation multipoles are included in the analysis. 5) For the resonant excited states, the predominant dipole radiation emitted by the charged oscillator is balanced by the energy gained from higher frequencies of classical electromagnetic zero-point radiation corresponding to odd multiples of omega-zero. 6) The Bohr frequency condition — the energy change equals Planck’s constant times omega-zero — appears naturally in a classical context. The presence of the Lorentz-invariant zero-point radiation seems completely hidden in providing the energy balance for the oscillating charged particle. The present analysis puts a classical electromagnetic understanding under the old quantum picture of electrons in classical orbits, but requires resonance of the mechanical system in classical electromagnetic zero-point radiation.

I. Introduction

A. Summary

In this article, we treat a nonrelativistic classical point charge e in a one-dimensional classical linear oscillator potential interacting with classical electromagnetic zero-point radiation. We show that the oscillating charge in the small-amplitude limit has a stable ground state and unstable resonant excited states which are analogous to those of the quantum harmonic oscillator. The present analysis provides a classical electromagnetic understanding under the old-quantum picture of electrons in classical orbits.

B. Basic idea

The purely nonrelativistic classical mechanical harmonic oscillator system with parameters of mass M and frequency omega-zero can be written in terms of action-angle variables, but there is no scale for the oscillator’s action variable. On the other hand, random classical electromagnetic zero-point radiation has exactly one scale, namely Planck’s constant, associated with the stochastic radiation action variable for each radiation normal mode where the average value is half Planck’s constant, and the stochastic process is independent for each radiation normal mode. Equilibrium between the charge and the zero-point radiation requires that the mechanical oscillator receives the average value for its action variable from the zero-point radiation. The action variable for the nth resonant excited state of the mechanical system equals an odd multiple of the radiation action variable, namely two n plus one times it. Energy balance between the mechanical system and the classical zero-point radiation is arranged by differences in the frequencies of the driving radiation and the oscillator motion: the driving frequency for the nth state is two n plus one times the natural oscillator frequency.

C. Outline of the article

After reviewing some basic background material, we develop six fundamental themes. 1) Classical electromagnetic zero-point radiation is Lorentz invariant and requires that any charged mechanical system which comes to equilibrium in it is at least approximately Lorentz invariant. Although the linear harmonic oscillator is allowed as approximately Lorentz invariant, very few other potentials are allowed. 2) We go beyond the dipole approximation and require that the interaction of relativistic electromagnetic waves with a low-velocity charge at position r(t) be treated as the field evaluated at r(t), not simply in the dipole approximation corresponding to the field at the origin. 3) The charged mechanical linear oscillator has SO(2) symmetry associated with the time behavior of the angle variable, and the integer values for resonances arise from the integer-indexed representations of the SO(2) symmetry group. 4) In the large-mass approximation, the ground state is stable even when higher radiation multipoles are included. 5) For the resonant excited states, the predominant dipole radiation emitted by the charged oscillator is balanced by the energy picked up from higher frequencies of classical electromagnetic zero-point radiation. 6) The Bohr frequency condition on transitions between resonant excited states appears naturally in the classical analysis. The presence of the Lorentz-invariant zero-point radiation seems completely hidden in providing the energy balance for the oscillating charged particle.

II. Background material

A. Electromagnetic model

The classical one-dimensional charged harmonic oscillator can be considered to arise in the classical electromagnetic model of the one-dimensional motion of a charge e between two fixed charges of the same sign as e. Then the moving charge e oscillates back and forth, being repelled by both of the fixed charges. Since the oscillating charge is oscillating and hence is accelerating, it loses energy through the emission of radiation. The charge e would eventually come to rest after having lost all its mechanical energy if it were not for the ambient classical zero-point radiation which provides random forces to accelerate the charge e. We want to consider the behavior of the oscillating charge in the classical zero-point radiation spectrum. We will choose the charged oscillator as oriented along the z-axis so that the spherical angle is zero, and there is no need to discuss the azimuthal angle.

B. Lorentz invariance, at least in approximation

The fundamental constants of electromagnetic theory are the elementary charge e, the speed of light c, and Planck’s constant. Thus there is room for one unit mass, which will set the scale for the theory. The unit of length can be taken as the charge squared divided by the unit mass times the speed of light squared, the unit of time as the charge squared divided by the unit mass times the speed of light cubed, and the unit of energy as the unit mass times the speed of light squared. The fine structure constant is the fundamental constant independent of the choice of unit mass. In the present article, the speed of light does not enter the mechanical motion of the low-velocity charge e, and hence the only appearance of the speed of light is in the connection of the charged mechanical system to radiation. Therefore the fine structure constant does not appear in the analysis. Also, the charge e appears for both the energy lost and the energy gained by the oscillator, and so cancels out completely in the equations for oscillator energy balance. In the nonrelativistic analysis, the constant c does not appear in the mechanical motion, and the charge e can assume any small value.

C. Classical electrodynamics with classical zero-point radiation

The existence of Casimir forces between uncharged conducting surfaces implies, within classical physics, that there must be random radiation even at the zero of temperature, termed classical electromagnetic zero-point radiation. The magnitude and distance behavior of the Casimir forces can be accurately accounted for by temperature-independent random classical radiation with a spectrum per normal mode given by an average energy of half Planck’s constant times the frequency, which is the spectrum of Lorentz-invariant classical zero-point radiation. We emphasize that the instantaneous radiation energy per normal mode is the action variable times the frequency, where the action variable is a stochastic process. The action variable for classical zero-point radiation has the same average value of half Planck’s constant for radiation of every frequency in every inertial frame.

Classical electrodynamics with classical electromagnetic zero-point radiation is a specific version of classical electrodynamics where the source-free solution of Maxwell’s equations is chosen as random classical radiation fields with a Lorentz-invariant energy spectrum. The theory is often termed stochastic electrodynamics. The scale of the classical zero-point radiation is chosen so as to give correctly the Casimir forces between conducting parallel plates. This classical theory contains Planck’s constant as the one and only scale for classical zero-point radiation. For over 50 years, the implications of this classical theory have been gradually obtained. The classical theory has given a number of results which are usually claimed to require quantum theory. For example, there are classical calculations for Casimir forces, van der Waals force, oscillator specific heats, diamagnetism, superfluid behavior, the absence of atomic collapse, and the blackbody spectrum which agree for average values with the results of experiment and with the corresponding quantum calculations.

D. Approximately relativistic behavior for large oscillator mass

Classical electromagnetic zero-point radiation is Lorentz invariant. Any nonrelativistic system, such as a nonlinear charged harmonic oscillator with arbitrary oscillation speed, will tend to push the Lorentz-invariant zero-point spectrum toward the Rayleigh-Jeans spectrum. However, a nonlinear oscillator is not even approximately Lorentz invariant because it contains a length parameter in the nonlinear term which can be combined with the natural oscillator frequency to give a parameter-based velocity which is different from the speed of light. For approximate Lorentz symmetry, we turn here to a linear oscillator which involves parameters of a mass and a frequency. Furthermore, it is only when the charged oscillator amplitude — depending on the oscillator’s action variable — is very small, and therefore its maximum speed is very small compared with the speed of light, that the linear oscillator motion is approximately in agreement with a relativistic system. Thus, we will insist that the motion of the charged particle e is small by taking its mass as very large so that the maximum oscillator kinetic energy is small compared with the rest energy, giving a speed far below the speed of light. The emphasis on relativity sharply restricts the mechanical systems to which the theory applies. The one-dimensional linear harmonic oscillator and the Coulomb potential are the most important allowed mechanical potential systems.

E. Oscillator ground state in the present analysis

It is important to realize that, when treated in the dipole approximation, the charged linear oscillator comes to steady state with any spectrum of random classical radiation, as does a nonrelativistic charged particle in a Coulomb potential. However, the assumption that the oscillator, in the limiting small-amplitude oscillation, comes to equilibrium for both its dipole and also its quadrupole radiation fields fixes the equilibrium spectrum, up to an overall constant, as Lorentz-invariant classical zero-point radiation. In its ground state, the interaction of the charged one-dimensional linear harmonic oscillator with radiation is completely disguised. The oscillator scatters the zero-point radiation, but there is no time-average net radiation propagating in any direction. One might be tempted to conclude that the charged harmonic oscillator had no interaction with radiation despite its continued oscillation.

F. Resonant excited states in the present analysis

All frequencies are present in random classical zero-point radiation. On the other hand, the oscillating mechanical system has only one natural oscillating frequency. If the charged mechanical oscillator is in its ground state in the presence of classical zero-point radiation, the radiation at both the dipole and quadrupole frequencies gives rise to the same stochastic process for the mechanical oscillator. An observer measuring radiation would be unaware of the radiation continually emitted and absorbed by the charged mechanical oscillator.

However, if the mechanical oscillator is well above the amplitude of the ground state, the charged oscillator is still emitting radiation, predominantly into the dipole radiation mode at the same frequency as its natural mechanical oscillation frequency. If the oscillator amplitude is correct for resonance with some driving zero-point radiation mode, the dipole radiation emission will be balanced against the dipole energy gained from zero-point radiation at a frequency different from the natural oscillation frequency of the mechanical oscillator. For these resonant excited states, the dipole radiation emitted by the charged oscillator at its fundamental frequency, with stochastic action variable equal to two n plus one times the radiation action variable, can be balanced by the dipole energy gain from zero-point radiation modes at a frequency equal to two n plus one times the natural mechanical oscillation frequency. However, the energy of the oscillator is the same as the energy of the associated zero-point radiation. On changes of resonant states, the situation leads to Bohr’s condition: the energy change equals Planck’s constant times the natural frequency. In the remainder of this article, we will carry out the calculations to confirm these statements.

III. The linear oscillator system

The one-dimensional classical linear harmonic oscillator oriented along the z-axis has a Hamiltonian given by the usual sum of kinetic and potential terms, where the position and the linear momentum are the dynamical variables on phase space, and the mass and the oscillation angular frequency are fixed parameters. The Hamiltonian may be rewritten in terms of action-angle variables as simply the action variable times the frequency, where the angle variable does not appear, nor does the mass. The solution of the equation of motion from this Hamiltonian, written as a multiply periodic expansion, involves absolutely no harmonics of the mechanical oscillation frequency.

(The remainder of Sec. III — the amplitude in terms of the action variable, and the equations of motion including the radiation reaction — is omitted for length; the complete text is at the source.)

IV. The spherical mode expansion for random classical radiation

The discrete integer values for certain parameters are associated with the irreducible representations of groups of symmetries. In previous analyses of classical electrodynamics with classical electromagnetic zero-point radiation, plane waves were used, which fitted with the emphasis on Lorentz invariance. For the random classical electromagnetic radiation in the present article, we will emphasize the rotational symmetry involving the rotation groups SO(2) and SO(3). Since the electromagnetic waves are in three spatial dimensions, we expand in terms of spherical multipole radiation fields, with random radiation in a very large spherical cavity of radius R written as a sum over magnetic and electric multipole fields with random phases and stochastic amplitudes.

(The remainder of Sec. IV — the explicit multipole expansion, the vector spherical harmonics, the normalization of the random amplitudes and the passage to the continuum limit — is omitted for length; the complete text is at the source.)

V. Energy absorbed from random ambient radiation

If the classical charged oscillator were located in absolutely empty infinite space, it would radiate away its mechanical energy and come to rest. In the presence of random classical radiation the emitted energy is balanced against the energy absorbed from the ambient field, and the balance must hold separately for each radiation multipole if the equilibrium is to be stable.

(The remainder of Sec. V — the energy balance for both dipole and quadrupole radiation, the treatment beyond the dipole-only approximation, the variation of parameters, the solution of the differential equation, and the energy gained from random classical radiation — together with Sec. VI, the ground state of the charged harmonic oscillator in zero-point radiation, and Sec. VII, the quadrupole radiation emission calculation, are omitted for length; the complete text is at the source.)

VIII. Resonant excited states

A. Preliminary ideas

1. Suggestive work by Huang and Batelaan and by Cole. In 2015, Huang and Batelaan showed that a classical charged harmonic oscillator in classical zero-point radiation would absorb radiation energy from a transient electromagnetic pulse. The absorption was at integer multiples of the natural mechanical oscillation frequency. Furthermore, in 2018, Cole pointed out that for a charged particle in a Coulomb potential, there were large resonances for driving radiation at integer multiples of the mechanical frequency of orbital motion. There were also resonant eccentricities of the orbital motion. Cole’s resonances corresponded to absorption of both energy and angular momentum by the charged particle sufficient to balance the loss of mechanical energy and angular momentum due to radiation emission. However, Cole did not consider driving by zero-point radiation, but rather treated driving by a circularly polarized plane wave incident normal to the orbital plane of the charged particle in the Coulomb potential, and of various wave amplitudes. This earlier work suggests that, for a charged mechanical oscillator in electromagnetic radiation, one might look for resonances in the spherical multipole radiation at integer multiples of the mechanical oscillation frequency.

2. Resonance for electromagnetic waves. A mechanical oscillator has exactly one resonant frequency, that of its natural frequency of oscillation. Thus when pushing a swing, there is only one resonant frequency. However, electromagnetic radiation is not limited to the fundamental oscillator frequency for the charged harmonic oscillator because of the position-dependence of the driving electric field, which must be evaluated at the moving charge. Classical zero-point radiation provides driving forces at all frequencies and all spherical multipoles. The presence of the charge’s own displacement in the argument of the radiation field means that the oscillator equation of motion corresponds to a parametric oscillator with an entirely new set of resonances at integer multiples of its natural oscillation frequency. The position-dependence of the driving electromagnetic field is an aspect which sometimes seems overlooked when considering resonant excited states.

3. Dipole behavior for resonant excited states. In the small-source, large-speed-of-light approximation appropriate for charged-linear-oscillator mechanical motion, the radiation from the spherical multipole moments of order l goes as one over the speed of light to the power two l plus one. For example, the dipole radiation involves power radiated as one over the speed of light cubed, whereas quadrupole radiation power goes as one over the speed of light to the fifth. Indeed, when the mechanical speed is small, the lowest possible value of l always dominates the radiation energy loss due to emission. For a point charge, the radiation power going as one over the speed of light cubed always dominates. Numerical integration of the energy-balance expressions regarded as functions of the zero-point driving frequency suggests that equally-spaced peaks in the mechanical motion appear when the driving frequency is an odd-integral multiple of the natural frequency, and the peaks beyond the first all involve about the same height and width. Accordingly, we will look for resonant excited states involving dipole oscillator emission when the radiation absorption is close to an odd integer multiple of the natural frequency.

(Sections VIII B and VIII C — the spherical Bessel function analysis for dipole radiation and the worked examples — are omitted for length; the complete text is at the source.)

IX. Energy transitions and Bohr’s rule

Even when in a resonant excited state, the loss of energy by the oscillator is mainly at the dipole frequency, but the energy gain is at the higher frequency equal to two n plus one times the natural frequency. There is no imbalance in the oscillator’s dipole energy during the time when the oscillator is in the resonant excited state. However, the higher driving multipoles are, in general, not in energy balance unless the oscillator is in its ground state. The transition from one value of n to another is associated with a change in the amplitude of the charged mechanical oscillator and also in the frequency of the driving radiation. If the integer n changes by one unit, then the change in average energy of the charged mechanical oscillator is exactly Planck’s constant times the natural frequency — Eq. (59) — and the frequency of the transition is the natural oscillation frequency of the charged oscillator in empty space.

The difference in energies of the driving radiation matches the change in mechanical energy of the oscillator. In transitions between different values of n, the change in energy for the oscillator is the same as the change in energy for the driving radiation. In the situation of oscillator energy balance at each resonance excited state labeled by n, it may appear as though the charged particle were not radiating at all, since the oscillator’s energy does not change. Net radiation appears only on changes of the index n. During the transition, the charged particle radiation is not balanced by the driving zero-point radiation. The energy change satisfies Bohr’s relation for the oscillator. Once again, just as for the ground state, the stabilizing role of the classical zero-point radiation in resonant excited states may seem completely hidden.

X. Stochastic processes versus eigenvalues

In our classical electrodynamic analysis, the random classical zero-point radiation is a stochastic process for each normal mode of frequency omega, with Hamiltonian equal to the action variable times the frequency. Thus, since the frequency is fixed for the normal mode while the action variable is a stochastic process, the energy for each normal mode must also be a stochastic process. The average value for the radiation action variable is half Planck’s constant, independent of the frequency. This stochastic process is then transferred to the charged mechanical oscillator, whose Hamiltonian is its action variable times its natural frequency and whose average energy is half Planck’s constant times that frequency.

This classical description is in contrast with the quantum viewpoint which regards the oscillator energy for any oscillator as an eigenvalue with no dispersion, equal to half Planck’s constant times the frequency. The average value of the classical analysis agrees with the expectation value of the quantum analysis, but both the description and the dispersion are completely different.

In both the classical and the quantum theories, the SO(2) symmetry involving position and momentum is recognized by some authors. However, in elementary quantum texts, the symmetry is often not acknowledged. The SO(2) symmetry involving position and momentum alone leads to the integer-indexed representations and to the average values. The fluctuations, however, are very different between the classical and quantum theories.

In any case, it seems comforting to those with classical sensibilities that some parts of old quantum theory can be understood as classical charges moving in classical trajectories under the fluctuations of random classical zero-point radiation.

XI. Concluding remarks

A. Resonance: its absence in classical statistical mechanics and its importance in classical electromagnetism

At the end of the 19th century and beginning of the 20th, there were repeated attempts to apply nonrelativistic classical statistical mechanics to phenomena associated with atomic physics. The Rayleigh-Jeans law is the result of such an attempt. Indeed, Planck investigated blackbody radiation from the perspective of thermodynamics, and connected the average energy of a charged harmonic oscillator with the average energy of the classical radiation modes at the same frequency as the harmonic oscillator. However, he did not introduce classical electromagnetic zero-point radiation.

There is a huge difference between the Brownian motion treated in texts of statistical mechanics and the motion of a charged mechanical system in classical electrodynamics with classical electromagnetic zero-point radiation. For example, due to collisions, a neutral, uncharged one-dimensional mechanical harmonic oscillator will come to thermal equilibrium at an energy set by the temperature alone, independent of the oscillator frequency. On the other hand, in electromagnetism, the response of a charged oscillator is resonant at its natural frequency. Indeed, the forcing of a charged oscillator by an electric field involves the location of the charged particle, so that the amplitude of the oscillation can influence the driving electromagnetic force on the oscillator. In the ground state, the higher multipoles of the classical zero-point radiation field lead to the same stochastic behavior for the charged oscillator as given by the dipole approximation. There is no change in the spectrum of classical zero-point radiation due to the charged harmonic oscillator in the small-source approximation. However, the dependence of the electromagnetic force on the amplitude of the oscillation will lead to a parametric forcing which leads to resonant excited states for a charged one-dimensional linear oscillator. In this article, we have used classical electrodynamics with classical electromagnetic zero-point radiation to describe the motion of a charged harmonic oscillator in zero-point radiation.

B. The ground state in zero-point radiation

For the one-dimensional charged harmonic oscillator ground state involving dipole radiation, the dipole radiation determines the amplitude of mechanical oscillation in terms of any spectrum of random radiation. The charged harmonic oscillator will come to equilibrium in any arbitrary spectrum of random radiation. On the other hand, if we require that classical electrodynamics holds and the radiation spectrum is in equilibrium for both dipole and quadrupole radiation, and any velocity-dependent damping for the oscillator is omitted, then, up to an overall multiplicative constant, the only allowed spectrum of random radiation is that of Lorentz-invariant classical zero-point radiation. In classical electromagnetic zero-point radiation, both the dipole and quadrupole radiation balance involve the same stochastic process for the charged oscillator. However, the quadrupole radiation is suppressed by additional powers of the speed of light. Based upon the ground state radiation behavior, one would be unaware of the presence of zero-point radiation for a charged harmonic oscillator in its ground state.

C. Resonant excited states and dipole radiation

For any resonant excited states, we do not expect all the radiation modes to contribute to the same stochastic process for the charged oscillator since excited states are unstable and decay. Therefore we focus our attention on the dipole radiation associated with possible excited states. In the nonrelativistic calculation for the charged mechanical system, dipole radiation is the predominant radiation multipole. Since zero-point radiation is Lorentz-invariant while the harmonic oscillator potential is not, compatibility requires that the velocity, and hence any amplitude of the oscillation, is very small. Now there is only one steady-state resonant frequency for the mechanical harmonic oscillator, namely its natural oscillation frequency. However, because of the position dependence of the driving radiation, and hence of the driving force, the resonant excited states involve basically the dipole spherical multipole fields, but for different frequencies between the emitted radiation and the driving radiation. For the one-dimensional charged harmonic oscillator, the radiation emission is always at the natural oscillation frequency, but, for resonant excited states, the oscillation amplitude is larger, whereas the driving radiation will be at the higher resonant frequency. What we require is that the net driving force contained in the field at the charge’s position contains a frequency component agreeing with the natural frequency of the mechanical oscillator.

We find that the oscillator energy of the charged oscillator, oscillating at its natural frequency, is the same as the energy of the driving zero-point radiation mode, whose frequency is two n plus one times the natural frequency. On change of the excited state, the energy change of the oscillator is the same as the energy change of the radiation modes, but the emitted radiation corresponds to that of a charged oscillator in empty space. The important role of classical zero-point radiation in setting the exact average amplitudes for both the ground state and the resonant excited states seems hidden. The present work provides a classical electromagnetic understanding for the old-quantum picture of electrons in classical orbits.

XII. Acknowledgements

I am deeply indebted to the work of Professor Daniel C. Cole whose article with Y. Zou kept the analysis of classical zero-point radiation advancing when only linear systems seemed successful, and whose work on subharmonic resonances was quite thought-provoking. The work by Professor Herman Batelaan and W. Huang was also intriguing. Furthermore, there are many books and articles on classical electrodynamics, quantum mechanics, quantum field theory, group theory, and statistical physics which have been instrumental in my understanding of the classical electromagnetic situation. I also wish to thank Professor Cole for noting typos and ambiguous aspects of the manuscript.

(The 29-item bibliography is omitted; the complete list is at the source.)