Stochastic electrodynamics (SED) predicts a Gaussian probability distribution for a classical harmonic oscillator in the vacuum field. This probability distribution is identical to that of the ground state quantum harmonic oscillator. Thus, the Heisenberg minimum uncertainty relation is recovered in SED. To understand the dynamics that give rise to the uncertainty relation and the Gaussian probability distribution, we perform a numerical simulation and follow the motion of the oscillator. The dynamical information obtained through the simulation provides insight to the connection between the classic double-peak probability distribution and the Gaussian probability distribution. A main objective for SED research is to establish to what extent the results of quantum mechanics can be obtained. The present simulation method can be applied to other physical systems, and it may assist in evaluating the validity range of SED.

According to quantum electrodynamics, the vacuum is not a tranquil place. A background electromagnetic field, called the electromagnetic vacuum field, is always present, independent of any external electromagnetic source [

Despite that the classical mechanics and SED are both theories that give trajectories of particles, the probability distributions of the harmonic oscillator in both theories are very different. In a study of the harmonic oscillator, Boyer showed that the moments

A comparison between the harmonic oscillators with and without the vacuum field. Top: without any external force except for the vacuum field, the SED harmonic oscillator undergoes a motion that results in a Gaussian probability distribution. This motion is investigated with our simulation. Bottom: in the absence of the vacuum field or any external drive, a harmonic oscillator that is initially displaced from equilibrium performs a simple harmonic oscillation with constant oscillation amplitude. The resulting probability distribution has peaks at the two turning points.

Although the analytical solution of an SED harmonic oscillator was given a long time ago, it is not straightforward to see the dynamical properties of a single particle from the complicated solution. Additionally, many results in SED such as probability distribution are obtained from (ensemble) phase averaging. Thus, they cannot be used for the interpretation of a single particle’s dynamical behavior in time (some may consider applying the central limit theorem and treat the positions in the time sequence as independent random variables. However, for an SED system this cannot work, because the correlation between the motion at two points in time persists beyond many cycles of oscillation) unless the system is proved to be ergodic. As an analytical proof of the ergodicity is very difficult, we take a numerical approach to study the particle’s dynamical behavior. Besides what is already known from the analytical solution, our numerical studies construct the probability distribution from a single particle’s trajectory. We investigate the relation between such a probability distribution and the particle’s dynamical behavior. Ultimately, we want to know the underlying mechanism that turns the classic double-peak distribution into the Gaussian distribution.

While most works in the field of SED are analytical, numerical studies are rare [

The major challenge for the numerical simulation is to properly account for the vacuum field modes. A representative sampling of the modes is thus the key for successful simulations. In this study, one of our goals is to use a simple physical system, namely, the simple harmonic oscillator, to benchmark our numerical method of vacuum mode selection so that it can be used to test the validity range of SED as discussed in the following.

Over the decades, SED has been criticized for several drawbacks [

Meanwhile, a modified theory called stochastic electrodynamics with spin (SEDS) was recently proposed [

The organization of this paper is the following. First, in Section

In his 1975 papers [

Given the knowledge of the moments

While the vacuum field in unbounded space is not subject to any boundary condition and thus every wave vector

Since the range of the allowed wave vectors

In the simulation, the summation indices in (

In the limit of

Finally, for a complete specification of the vacuum field, (

The isotropic distribution of the polarization field vectors

In summary, the vacuum field mode

In the unbounded (free) space, the equation of motion in Boyer’s analysis is

As an additional note, given the estimation of

Finally, as we have established an approximated equation of motion (

To summarize, (

In Section

To construct the probability distribution from a particle’s trajectory, two sampling methods are used. The first method is sequential sampling and the second method is ensemble sampling. In sequential sampling the position or velocity is recorded in a time sequence from a single particle’s trajectory, while in ensemble sampling the same is recorded only at the end of the simulation from an ensemble of particle trajectories. The recorded positions or velocities are collected in histogram and then converted to a probability distribution for comparison to the analytical result (

By solving (

A comparison between particle trajectory and the temporal evolution of the vacuum field. Top: the vacuum field (red) is compared to the trajectory of the SED harmonic oscillator (black). Bottom: a magnified section of the trajectory shows that there is no fixed phase or amplitude relation between the particle trajectory and the instantaneous driving field. The modulation time for the field is also shown to be longer than that for the motion of the harmonic oscillator.

Here we would like to highlight some interesting features of the simulated trajectory (Figure

The sequential sampling of a simulated trajectory gives the probability distributions in Figure

The probability distribution constructed from a single particle’s trajectory (number of sampled frequencies

To understand how the trajectory gives rise to a Gaussian probability distribution, we investigate the particle dynamics at two time scales. At short time scale, the particle oscillates in a harmonic motion. The oscillation amplitude is constant, and the period is

Contributions of different oscillation amplitudes in the final probability distribution. Top: several sections (red) of a steady-state trajectory (black) are shown. A section is limited to the duration of the characteristic modulation time. The oscillation amplitude changes significantly beyond the characteristic modulation time, so different sections of the trajectory obtain different oscillation amplitudes. Bottom: the probability distributions of each section of the trajectory are shown. As the oscillation amplitude is approximately constant in each section, the corresponding probability distribution (red bar) is close to the classic double-peak distribution. The probability distributions in different sections of the trajectory contribute to different areas of the final probability distribution (black dashed line). The final probability distribution is constructed from the steady-state trajectory.

A schematic illustration of the oscillation amplitude as a sum of different frequency components in the complex plane. At a particular time

A representative sampling of the oscillation amplitudes with each sampled amplitude separated by

Sampled oscillation amplitude and the reconstructed Gaussian probability distribution. Top: the oscillation amplitudes (red dot) are sampled from a steady-state trajectory (black) with a sampling time-step equal to

In many SED analyses [

The ensemble sampling of the simulation gives the probability distributions in Figure

The probability distribution constructed from an ensemble sampling (number of particles

Radiation damping and Heisenberg’s minimum uncertainty relation. Under the balance between vacuum field driving and the radiation damping, the trajectory of the SED harmonic oscillator (black) satisfies the Heisenberg minimum uncertainty relation. When the radiation damping is turned off in the simulation (red and blue), the minimum uncertainty relation no longer holds, although the range of the particle’s motion is still bounded by the harmonic potential.

Unlike sequential sampling, ensemble sampling has the advantage that the recorded data are fully uncorrelated. As a result, the integration time does not need to be very long compared to the coherence time

The inverse relation between the computation time and the number of processors. The data (black open square, blue, and red dot) follow an inverse relation

The ensemble-averaged energy of the SED harmonic oscillator and its convergence as a function of sampled frequency number

The analytical probability distribution of an SED harmonic oscillator is obtained in Section

As the probability distribution constructed from a single trajectory is a Gaussian and satisfies the Heisenberg minimum uncertainty relation, we investigate the relation between the Gaussian probability distribution and the particle’s dynamical properties. As a result, the amplitude modulation of the SED harmonic oscillator at the time scale of

A comparison between the harmonic oscillators with and without the vacuum field. Top: compared to Figure

In quantum mechanics, the harmonic oscillator has excited, coherent, and squeezed states. A natural extension of our current work is to search for the SED correspondence of such states. Currently, we are investigating how a Gaussian pulse with different harmonics of

The methods of our numerical simulation may be applicable to study other quantum systems that are related to the harmonic oscillator, such as a charged particle in a uniform magnetic field and the anharmonic oscillator [

Lastly, over the last decades there has been a sustained interest to explain the origin of electron spin and the mechanism behind the electron double-slit diffraction with SED [

On the other hand, over the years claims have been made that SED can predict double-slit electron diffraction [

Schematic illustration of a vacuum field based mechanism for electron double-slit diffraction. Several authors have proposed different SED mechanisms that explain the electron double-slit diffraction. The central idea is that the vacuum field in one slit is affected by the presence of the other slit. As the vacuum field perturbs the electron’s motion, an electron passing through only one slit can demonstrate a dynamical behavior that reflects the presence of both slits. Such a mechanism may reconcile the superposition principle with the concept of particle trajectory.

In “unbounded” space, the modes are continuous and the field is expressed in terms of an integral. In “bounded” space, the modes are discrete and the field is expressed in terms of a summation. In both cases, the expression for the field amplitude needs to be obtained (see Appendices

The homogeneous solution of Maxwell’s equations in unbounded space is equivalent to the solution for a wave equation:

Without loss of generality, a random phase

The solution of homogeneous Maxwell’s equations in bounded space has the summation form:

Using the relation (if the two modes are not identical (i.e.,

A wave vector chosen along the

A field composed of finite discrete frequencies,

Repetition time of a beat wave. A beat wave (black solid line) is made of two frequency components (red and blue solid lines). The oscillation periods of the two frequency components are

This work was completed utilizing the Holland Computing Center of the University of Nebraska. In particular, the authors thank Dr. Adam Caprez, Charles Nugent, Praga Angu, and Stephen Mkandawire for their work on parallelizing the SED simulation code. Huang thanks Dr. Bradley A. Shadwick, Roger Bach, and Dr. Steven R. Dunbar for stimulating discussions. This work is supported by NSF Grant no. 0969506.