Two New Methods in Stochastic Electrodynamics for Analyzing the Simple Harmonic Oscillator and Possible Extension to Hydrogen

Abstract

The position probability density function is calculated for a classical electric dipole harmonic oscillator bathed in zero-point plus Planckian electromagnetic fields, as considered in the physical theory of stochastic electrodynamics (SED). The calculations are carried out via two new methods. They start from a general probability density expression involving the formal integration over all probabilistic values of the Fourier coefficients describing the stochastic radiation fields. The first approach explicitly carries out all these integrations; the second approach shows that this general probability density expression satisfies a partial differential equation that is readily solved. After carrying out these two fairly long analyses and contrasting them, some examples are provided for extending this approach to quantities other than position, such as the joint probability density distribution for positions at different times, and for position and momentum. This article concludes by discussing the application of this general probability density expression to a system of great interest in SED, namely, the classical model of hydrogen.

Keywords: stochastic electrodynamics; classical physical dynamics; hydrogen; harmonic oscillator; nonlinear dynamics.

1. Introduction

This paper involves the physics of stochastic electrodynamics (SED) and the exploration of a new approach for analyzing probabilities associated with charged particle motion due to the interaction with stochastic electromagnetic radiation. SED involves the movement of classical charged point particles while interacting with a specific form of fluctuating classical electromagnetic radiation. Despite SED being completely classical, agreement has been shown between SED and quantum mechanics (QM) and even quantum electrodynamics (QED), under an interesting range of conditions. The "classical physics" aspects of SED consist of electromagnetic radiation that obeys Maxwell's classical, microscopic electromagnetic equations, while the classical charged particles obey the relativistic Lorentz–Dirac classical equation of motion. The physical predictions of SED that agree with QM and QED hold for classical systems with a linear differential equation of motion. A very good demonstration of this point is Boyer's work for the simple harmonic oscillator (SHO). Even the complicated fully retarded van der Waals forces between atoms modelled by electric dipole oscillators fulfill this agreement, as do Casimir forces between continuum materials; this agreement holds in both cases for all temperature conditions. Interestingly enough, at one point many who have studied and explored SED thought that SED might form the basis for QM and QED. However, complications have since been found to persuade most researchers that this is not the case.

The disagreement between SED and QM for all arbitrary "atomic systems" has some bearing for motivating the investigation in this paper. An interesting point to explain why all "atomic systems" covered by QM do not also hold for SED was first made and analyzed by Boyer and subsequently followed up in a different way by the present author. The point was this: the binding force for all atomic systems in nature is due to the Coulombic force. Hence, not just any binding force inserted into the SED description, other than the Coulombic-based one, should be expected to share agreement with real atomic systems in nature; moreover, SED should not be expected to agree with nonphysical "atomic systems" containing arbitrary binding forces in QM. The difficulty here is that such Coulombic-based systems are inherently nonlinear for the equation of motion for the classical electrons interacting with the classical nucleus; hence, such systems are far more complicated to analyze in SED than when the differential equation of motion is linear. More about this point is discussed in the concluding section of Section 4, but this forms much of the motivation in this study for examining a new way of calculating probabilities in SED.

Key aspects of SED concern classical charged particles interacting with classical electromagnetic radiation at some temperature T, with the recognition that at zero temperature the radiation is nonzero, with particular properties that enable a statistical equilibrium between charged particles and radiation. Some of the specific properties of classical electromagnetic zero-point (ZP) radiation at zero temperature include that the radiation frequency distribution must be Lorentz invariant and that the fundamental definition of zero temperature must be obeyed by ZP radiation. In SED, the zero-temperature stochastic radiation is referred to as ZP radiation. The stochastic radiation at and above zero temperature is referred to here as "ZP plus Planckian" (ZPP) radiation.

This study involves exploring the use of the expression given as equation 1 in the article: the probability density of finding a point charged particle at position x in the steady state is written as an integral over all the Fourier coefficients A1 through AN and B1 through BN of the radiation field, of the probability density of those coefficients multiplied by a three-dimensional Dirac delta function that picks out the particle's trajectory. The restriction to "steady state" can be removed, but that results in a time dependence which is not treated here. Other expressions beside the one for position, such as those for the energy and momentum, are discussed in Section 2.2.

The "3x" notation in equation 1 indicates that the function refers to the position vector point in three-dimensional space, while the "F,A-B" notation refers to the probability density function for the Fourier coefficients of the electromagnetic radiation described next. Specifically, the coefficients in equation 1 represent the coefficients in the Fourier expression for the electric and magnetic fields that the particle is "immersed" within. At the end of the calculations, the number of coefficients is taken to infinity. Finally, the trajectory appearing inside the delta function is the steady state trajectory of the particle and is a function of all the coefficients. All the coefficients are real quantities and are integrated over the whole real line. Their values control the steady state solution for the particle's trajectory. The probability density of these fields dictates their contribution to how the delta function "selects" the contribution to the final particle probability density.

Two solution methods are examined in this paper. In Section 2.1, a slight variation to equation 1 is used to fully evaluate the analytic probability density for the position of a one-dimensional (1D) SHO. Here, each Fourier coefficient is explicitly integrated over. In Section 2.2, a few other examples using expressions similar to equation 1 are also discussed for the 1D and 3D SHOs, including the particle's joint probability densities for position at two different times and for position and momentum, as well as the probability density for the kinetic plus potential energy of the oscillator. In Section 3, a second method, different from the direct integration method in Section 2.1, is described for deducing the analytic expression for the position probability density. This second method uses a partial differential equation (PDE) approach. Finally, Section 4 provides some concluding remarks, including a discussion of generalizing this study to more complicated systems, in particular the classical hydrogen atom.

2. Calculating analytic SHO probability density functions from initial general expression

2.1. Direct integration method for the analytic expression of the SHO probability density

The use of probability density expressions like equation 1 has been explored before, but mainly for the stochastic electric field values of the radiation fields. In this paper, the following is analyzed: the stochastic properties of a classical charged point electron, bound by an SHO potential and bathed in stochastic classical electromagnetic ZPP radiation.

To start, let us describe the radiation fields by considering a large region of space, where "large" means as compared to the size of the space that any charged particles, "bound" by a classical potential, occupy via traversing within the confines of this classical potential. Let us consider a rectangular parallelepiped region that the radiation fields are confined within, with dimensions Lx, Ly and Lz along the x, y and z axes. Although other shapes can be considered, the rectangular parallelepiped offers mathematical simplicity, especially since at the end of the calculations these dimensions are typically taken in the limit of infinite size.

In what follows, the ZP or ZPP radiation fields are represented as an infinite sum of plane waves, with periodic boundary conditions imposed. This imposition enables the use of Fourier series to describe the fields. For a large region of space, this periodicity does not affect the physical analysis, but it does simplify the subsequent mathematical analysis. Equations 2 and 3 of the article accordingly write the free electric and magnetic radiation fields as sums over wavevectors and over the two linear polarizations, each mode carrying a cosine term with coefficient A and a sine term with coefficient B; equation 4 gives the allowed wavevectors in terms of the box dimensions and three integers. The frequency of a mode is the speed of light times the magnitude of its wavevector, the polarization vectors are perpendicular to the wavevector and to each other. Using the free space Maxwell equations one can show that these expressions satisfy the wave equation for both fields, and the geometry of the polarization vectors ensures that all four free space Maxwell equations are satisfied.

In SED, the stochastic nature of these radiation fields arises from the probability distribution of the Fourier coefficients over a large ensemble of equally sized space regions. The fields within the ensemble of space regions are characterized by the temperature T; consequently, the Fourier coefficients will have a probabilistic distribution over the ensemble. For each member of the ensemble, the Fourier coefficients are fixed; only when examining each cavity in the ensemble will the Fourier coefficients be different and follow a probabilistic distribution in values.

Although this description is followed in SED, this basic behavior goes back to Planck in the first half of Planck's major treatise and later to Einstein and Hopf. The main difference between those much older studies and SED is that in SED the assumption is not made that the radiation fields fall to zero at zero temperature.

Here, a classical charged point particle with charge q is considered oscillating in one dimension, constrained along that direction by a simple harmonic oscillator potential and so subject to a linear binding force. If one imagines a sphere of uniform charge density and net charge minus q that the plus q point charge oscillates within, then the oscillator force acting on the point charge can be pictured as originating in this way. This neutral system will look like an electric dipole oscillator at distances far from the oscillator system.

As for the oscillating charge, one can describe its motion using the nonrelativistic approximation of the Lorentz–Dirac equation, equation 5 of the article: mass times acceleration equals the linear restoring force, plus the nonrelativistic radiation-reaction term proportional to the third time derivative of position with the coefficient tau equal to two thirds of the charge squared divided by the mass times the cube of the speed of light, plus the charge times the radiation electric field evaluated at the origin. The field is evaluated at the origin because the dipole approximation is being made when evaluating the electric field component of the Lorentz force. The magnetic field component of the Lorentz force is assumed to be much smaller in magnitude and is ignored here. A common approximation to the weak radiation-reaction term, due to the small magnitude of its coefficient, turns the third derivative into the first derivative times the square of the natural frequency, giving equation 7. Using equation 7 and the properties of the ZP and ZPP radiation fields, Boyer showed that a detailed agreement exists between SED on the one hand and QM and QED on the other, for the stochastic properties of this SHO system, at all temperatures.

Rewriting the field in the dipole approximation gives equation 8, and the steady state particular solution of the equation of motion is equation 9: a sum over modes in which each Fourier coefficient is divided by the familiar resonant denominator, the difference between the square of the natural frequency and the square of the mode frequency, plus an imaginary damping term. This expression is linear in the Fourier coefficients of the radiation electric field, and that linearity is what makes the whole calculation possible.

In SED, the Fourier coefficients are assumed to be independent random variables with a Gaussian probability density distribution, equation 14, whose width for each mode depends on the wavevector and on the temperature. This dependence has been studied considerably in SED. In particular, the functional form at zero temperature is a cornerstone of SED: the variance of each coefficient is Planck's reduced constant times the mode frequency divided by two pi. It is this functional form at zero temperature that was referred to in Section 1, deduced first via the imposition of Lorentz invariance by Marshall and by Boyer, and much later by the present author by imposing the thermodynamic definition of zero temperature.

The remainder of Section 2.1 carries out every one of these integrations in turn. (The chain of Fourier-coefficient integrals from equations 15 to 27 is omitted here because it cannot be rendered faithfully in plain text; the complete derivation is at the source.) The result is equation 28: the position probability density is a normalized Gaussian in x with variance sigma-x squared. This Gaussian result has been deduced in SED previously by researchers in SED, but not, as far as this author knows, starting via the general probability expression in equation 1. Although the above is a much longer derivation than deductions published earlier, it is still illuminating.

Moreover, while equation 28 is connected with QED, one can relate it to the relevant expression in QM as calculated from Schrödinger's equation when taking into account the probability density at temperature T, and summing over all the squared energy eigenfunctions, each weighted by its Boltzmann factor. To obtain this agreement, and essentially dropping the QED effects, in SED one would make what some call the continuum and resonant approximations, where first the sum over modes is approximated as a three-dimensional integral, and then later the charge is assumed to be small.

Carrying out those approximations, and using the fact that for an electron the radiation-reaction coefficient is very small — about 6.27 times ten to the minus twenty-four seconds — so that the resonant denominator is strongly peaked at the natural frequency, the mean square steady state displacement of the oscillator becomes equation 38: Planck's reduced constant divided by twice the mass times the natural frequency, multiplied by the hyperbolic cotangent of the quantum of energy divided by twice the thermal energy. At zero temperature this is equation 39: Planck's reduced constant divided by twice the mass times the natural frequency — the quantum ground-state value, reached here entirely within classical physics.

2.2. Examples of other analytic SHO probability density functions that can similarly be deduced

The method of Section 2.1 can be used to obtain many other types of probability density functions. Using the equation of motion in three dimensions, one can certainly generalize the previous one-dimensional oscillator to a three-dimensional one with the probability density of equation 1.

Moreover, the position and momentum joint probability density function for this three-dimensional oscillator can be expressed by equation 41, an integral over all the Fourier coefficients of the coefficient probability density multiplied by delta functions for both position and momentum, which can be used to find an analytic expression in a similar manner to Section 2.1. In addition, one can express the probability density for the nonrelativistic energy of a three-dimensional oscillator via equation 42, with the kinetic plus potential energy written out in equation 43.

As another example, using the same method, one can calculate the joint probability position density distribution for two different times. The final result, equation 44, is a two-dimensional Gaussian built from the mean square displacement and from the two-time correlation of the steady state solution. The ensemble average of the square of the steady state solution is independent of time, as given in the more familiar continuum and resonant approximation by equation 38. However, the ensemble average of the product of the positions at two times depends on the time difference, and is given by equation 45; in the continuum and resonant approximation this simplifies to equation 46, namely the mean square displacement multiplied by the cosine of the natural frequency times the time difference.

Thus, the general expressions are quite straightforward to formulate, as in equations 1, 16, 41, 42 and 44, although carrying out all the integrations to arrive at an analytic expression, as in equations 28 and 44, can be quite nontrivial.

3. A PDE approach for deducing the SHO probability density function

Rather than directly integrating over all the radiation Fourier coefficients in equation 16 to obtain the analytic expression for the position probability density in equation 28, here it is shown that equation 16 satisfies a PDE that enables equation 28 to be deduced. In some ways, this approach is less complicated than the direct integration method of Section 2.1 and might provide insight for more complicated systems than the oscillator, such as the classical hydrogen case.

Without integrating over all the coefficient variables as in Section 2.1, nor by only showing that equation 28 solves the PDE, here the one-dimensional version of equation 1 is taken directly and shown to satisfy equation 47: the derivative of the probability density with respect to the variance of position equals one half of the second derivative of the probability density with respect to position. This is the form of a diffusion equation in which the variance plays the role usually taken by time. More specifically, as is shown in the article, in order for equation 47 to hold with the probability density given by the one-dimensional version of equation 1, the variance must be given by equation 25. Moreover, upon imposing that the probability density tends to zero as the magnitude of the position grows without bound, with symmetry about the origin, and that the function monotonically decreases as the magnitude of the position increases, one obtains equation 28 as the solution of equation 47.

(The demonstration in Section 3 that the master expression satisfies this equation, equations 48 to 60 of the article, is a further chain of coefficient integrals and is omitted here for the same reason; the complete derivation is at the source. The Gaussian functional form of the coefficient probability density is what makes the demonstration go through, and it matters twice: once to secure the convergence of the integrals, and once to enable the squares to be completed.)

4. Concluding remarks

Two methods were shown in this paper for finding the analytic expression, equation 28, for the probability density of the position of a classical charged point particle in an SHO potential, where the charge is bathed in classical electromagnetic random radiation at a temperature T. Both of these methods used the one-dimensional form of equation 1 as the starting point. Section 2.1 obtained the analytic expression by explicitly integrating over each of the Fourier coefficients; the calculation was fairly lengthy. In contrast, Section 3 showed that the one-dimensional version of equation 1 satisfied a PDE, which in turn enabled equation 28 to be deduced.

Some other relevant points of this study are the following. First, equation 1 should hold for dynamic systems other than the SHO. Moreover, it should also hold for the SHO in more generality than considered here, namely, not just for the steady state part of the oscillatory motion, but also including the initial transitory motion. Including the probability density of the initial conditions in equation 1 would enable this to be accomplished. The probability density then becomes time dependent.

A dynamic system of considerable interest to be considered here is classical hydrogen, with a negatively charged classical point particle as the classical electron, and a much more massive nucleus with the opposite charge for the proton. In SED, this would again be "bathed" in classical electromagnetic random radiation at temperature T. This system is of interest as it represents a real atomic system, as opposed to the solvable, but hypothetical, SHO. Hydrogen is the simplest of atomic systems, so it is a suitable system to be analyzed in detail. Many researchers in SED have tackled this problem, but it still remains an open problem.

Equation 1 should hold for this classical hydrogen system. The trajectory inside the delta function would become the trajectory of the classical electron given its initial conditions and how the radiation fields influence the electron's trajectory, just as occurs for the SHO treated here. The equation of motion for the classical electron should be the Lorentz–Dirac equation, where the relativistic version is used, although it would be interesting to obtain and compare the resulting probability density when the nonrelativistic approximation of the Lorentz–Dirac equation is used. It should be noted that the "easiest" and clearest situation to be considered for this problem would be the zero-temperature case, since we expect that the electron would be bound and would not move off to infinity in space. Above zero temperature, there is a nonzero probability that the electron will "ionize" and travel arbitrarily far away, so this situation is more difficult to analyze with the present scheme.

Despite equation 1 being valid for this classical hydrogen atom, there is a significant difference in using this formulation for the classical hydrogen atom versus the SHO model analyzed here. The "success" of Sections 2.1 and 3 came about because of the following: a nonrelativistic equation of motion was used, the dipole approximation was made for the radiation's electric field, and the radiation magnetic field effects were ignored. These approximations enabled an analytic result for the motion to be obtained, equation 9. This expression is linear in the Fourier coefficients of the radiation's electric field. This linear analytic expression for the particle's motion was used in each of the methods, to arrive at the analytic expressions of equations 25, 26 and 28. Without the linear analytic expression of equation 9, the methods in Sections 2.1 and 3 could not have been carried out in the manners described.

In contrast, an analytic expression is not known for calculating the classical electron's probability distribution for the mentioned classical hydrogen atom, whether one treats the electron's trajectory relativistically, or even with a simpler nonrelativistic approximation. Although the expression in equation 1 should be correct for this classical hydrogen atom, other mathematical or numerical methods would need to be developed to carry out similar approaches to those in Sections 2.1 and 3, of direct integrations or showing the expression satisfies an appropriate PDE from which the probability density can be deduced.

To date, no researcher has found an analytic means within SED for deducing the position probability density for the classical hydrogen atom. Consequently, the author along with Y. Zou carried out simulation methods in 2003 for deducing it for hydrogen, with some degree of success. More recently, extensive simulations have been carried out by Nieuwenhuizen and Liska with interesting results, but always with some fraction of the ensemble of hydrogen systems resulting in electrons leaving, or "ionizing," away from the classical nucleus. Despite this problem, all three simulation efforts do not result in the classical electron "falling," or spiraling, into the nucleus due to energy radiating off from the classical electron's orbital motion. Thus, these simulations in SED "solve" the old atomic collapse problem of the simple classical atomic model by Rutherford.

Although these simulations are insightful, there are reasons for concern. Two of the simulations were not relativistic, while the third was certainly more relativistic than the others, but still not completely so, and each has various physical approximations. Perhaps of even more concern is that all of these studies deal with a chaotic system, where small errors in electron trajectories cannot of course be avoided numerically, but that build up to large errors quickly. Could these account for the apparent ionizations of some classical electrons in the ensemble of systems investigated? No matter how much the numerical resolutions of the simulations are reduced, this effect cannot go away, as known from chaos theory.

An analytic solution is indeed ideal for overcoming such problems, but may not be possible to obtain, whether nonrelativistically or relativistically. Nevertheless, this goal of exploring equation 1 for obtaining analytical results was part of the motivation for this study. What is interesting to note is that using Feynman's path integral method in QM was certainly exceptionally successful for a range of systems, but for a long time, starting from about 1948 until Duru's and Kleinert's paper in 1979, the hydrogen atom was not solved via this path integral method. Moreover, it should be mentioned that equation 1 has some resemblance to a "path integral" formulation, although the Fourier coefficients of the stochastic radiation field are integrated over instead of the possible paths of the particle.

Recapitulating, in this paper, the general expression of equation 1 was used to obtain an analytic expression for the position probability density of the one-dimensional electric dipole SHO, one of the first systems analyzed in SED. The calculations were fairly long for each of the two methods discussed, but certainly tractable. Despite equation 1 being correct for the classical hydrogen atom, evaluating equation 1 for this system is a far more complicated task for the reasons discussed.

Funding: This research received no external funding. Conflicts of Interest: The author declares no conflict of interest.

(The reference list is omitted for length; the complete text is at the source.)