# Discrete Excitation Spectrum of a Classical Harmonic Oscillator in Zero-Point Radiation

## Abstract

We report that upon excitation by a single pulse, a classical harmonic oscillator immersed in the classical electromagnetic zero-point radiation exhibits a discrete harmonic spectrum in agreement with that of its quantum counterpart. This result is interesting in view of the fact that the vacuum field is needed in the classical calculation to obtain the agreement.

### Keywords

Stochastic electrodynamics Quantum mechanics Classical dynamics Vacuum field Harmonic oscillator Excitation## 1 Introduction

Historically, the discreteness of atomic spectra motivated the early development of quantum mechanics. Modern quantum mechanics describes atomic spectra with high accuracy. Classical mechanics incorrectly predicts the radiation death of an atom and no discrete states. In the theory of stochastic electrodynamics (SED) classical mechanics is modified by adding a classical vacuum field. The vacuum field is called classical as it is a solution to Maxwell’s equations. The spectrum of this field is also required to exhibit Lorentz invariance [1, 2]. Planck’s constant \(\hbar \) is brought into the classical vacuum field as an overall factor that sets the field strength. SED avoids radiation death in that the electron reaches an energy balance between radiative decay and vacuum field absorption. This leads to the existence of a ground state. However, the prediction of discrete atomic spectra including excited states appears not to be within reach of SED. Indeed, Peter W. Milonni has commented in his well-known book *The Quantum Vacuum* that “Being a purely classical theory of radiation and matter, SED is unable... to account for the discrete energy levels of the interacting atoms” [3].

Some authors, do not share this view and have recently claimed that SED accounts for the behavior associated with quantized states [4]. Even if these claims for SED were correct, one may be tempted to ignore SED as a viable alternative to quantum mechanics altogether. After all, experimental tests of Bell’s inequalities establish that nature behaves in a nonlocal-real fashion, which appears to rule out any classical theory. The proponents of SED counter this idea by demonstrating entanglement-like properties [5]. In a broader view, SED is one of several attempts to build an “emergent quantum theory” [4].

In this paper we attempt to falsify SED with one counter example. We limit ourselves to the excitation process of an harmonic oscillator that is initially in the ground state. The results are compared to a fully quantum mechanical calculation. (Our original intent was to identify the limits of the validity range of SED before using it as an convenient means to study vacuum field effects on ground states). We find that SED can reproduce the discrete excitation spectrum of the harmonic oscillator, where the discreteness is explained as a result of parametric resonance (see detailed results and discussion below). This substantiates some claims by Cetto and gives credence to the idea that a full classical understanding of the Planck spectrum is possible [4] as it relies on the equidistant discrete spectrum of the harmonic oscillator. It is interesting and perhaps surprising that it is claimed that classical physics can explain physics phenomena that were long thought to be the exclusive domain of quantum mechanics. This list appears to include amongst others, the Black-Body Planck spectrum [4], the Casimir effect [6], and weak measurement [7].

We failed to provide a counter example to SED for the excitation spectrum of the harmonic oscillator. Nevertheless, the individual outcomes of measurements for SED may not agree with quantum mechanics. The quantum postulate for measurement states that individual outcomes for energy measurements can only take on energy eigenvalues, in agreement with observation. For the harmonic oscillator these outcomes are discrete, while the SED results in our study have a continuous distribution of energies for individual trajectories. It remains to be seen if SED can describe an individual measurement as the interaction between two systems (such as harmonic oscillators or atoms) that would mimic quantum mechanics and observation.

The organization of this paper is the following. First, the quantum and classical SED oscillator are considered. Their excitation spectra are obtained through numerically solving the equations of motion. A perturbative analysis is also given to provide insights to the underlying mechanism of the integer-spaced excitation spectrum of the classical SED oscillator.

## 2 Quantum Harmonic Oscillator

## 3 Classical Harmonic Oscillator in the Vacuum Field

## 4 Results and Mechanism

When the vacuum field is absent, the classical harmonic oscillator has only a single resonance at its natural frequency. With the vacuum field acting as a background perturbation, the classical harmonic oscillator exhibits an integer-spaced excitation spectrum, or if one likes, a “quantized” excitation spectrum. The position and the magnitude of the resonance peaks are in agreement with the quantum mechanical result. Such an agreement between the classical theory (as modified by the vacuum field) and quantum mechanics appears to be astonishing, given that the theory is fully classical and no quantization condition is added.

^{1}\(m = 9.11\times 10^{-35}~\mathrm{kg}\), polarization angle \(\theta _{p} = \pi /4\), pulse duration \(\Delta t= 10^{-14}~\mathrm{s}\), pulse center time \(t_{c} = 5\tau _{rel} = 1.60 \times 10^{-12}~\mathrm{s}\), and field amplitude \(A_{p} = 1.5\times 10^{-9}~\mathrm{Vs/m}\), the perturbation result shown in Eq. (37) gives the peak heights at \(1\omega _{0}\), \(2\omega _{0}\), and \(3\omega _{0}\) as

In summary, while the quantized excitation spectrum of a quantum harmonic oscillator is explained by the intrinsic quantized energy levels and the transitions associated with nonlinear operators \(\hat{x}^{n}\) in the excitation pulse, the “quantized” excitation of the classical harmonic oscillator is a result of parametric excitation due to the pulse and the initial conditions introduced by the vacuum field. In both theories, the integer-spaced overtones are caused by the nonlinearity of the pulse.

## 5 Discussion and Conclusions

We have shown that the classical harmonic oscillator in the vacuum field exhibits the same integer-spaced excitation spectrum as its quantum counterpart. This supports some of Cetto’s claims [4], and is especially interesting given the classical SED explanation that based on these claims can be provided for the black body spectrum. Our simulation is limited in the sense that it cannot resolve resonances at fractional frequencies (such as \(1/2\omega _{0}\) and \(1/3\omega _{0}\)). Such resonances are predicted by the quantum mechanical calculation, but are so weak that they are beyond the resolution of the SED simulation.

In this study, the classical and the quantum excitation spectrum are compared in terms of ensemble averages. The individual outcomes of measurements for SED appear not to agree with quantum mechanics. The quantum postulate for measurement states that individual outcomes for energy measurements can only take on energy eigenvalues, in agreement with observation. For the quantum harmonic oscillator these outcomes are discrete, while the classical theory of SED in our study gives a continuous distribution of energy outcomes for individual trajectories.

A question remains what a measurement constitutes in SED. If one would measure the frequency of the decay radiation, SED would likely predict this to be peaked around the natural frequency of the harmonic oscillator. It remains to be seen if the vacuum field can provide a mechanism to stimulate emission frequencies at the higher harmonics. Given that an excitation pulse can provide such a mechanism, it may be that the vacuum field will give rise to a similar decay spectrum to what quantum mechanics predicts. However, even if the SED decay spectrum would match the quantum mechanical spectrum, the amount of energy released in the decay will not match that of individual photons. The reason is that the radiated energy equals the amount of energy lost from the particle, which reflects its classical continuous spectrum.

Thus, in agreement with the nature of, and expanding on Milonni’s comment; although SED can account for at least one discrete energy spectrum in terms of averaged energies, it does (up to this point) not match quantum mechanics. It would be interesting to investigate if the SED description of two interacting systems would modify these results in bring them closer to the quantum mechanical predictions.

## Footnotes

- 1.
The mass value is chosen to keep the integration time manageable without losing the physical characteristics of the problem.

## Notes

### Acknowledgments

We gratefully acknowledge comments from Prof. Peter W. Milonni. The funding support comes from NSF Grant No. 0969506. This work was completed utilizing the Holland Computing Center of the University of Nebraska and the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF Grant No. OCI-1053575.

### References

- 1.Boyer, T.H.: Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation. Phys. Rev. D
**11**, 790 (1975)CrossRefADSGoogle Scholar - 2.Boyer, T.H.: General connection between random electrodynamics and quantum electrodynamics for free electromagnetic fields and for dipole oscillator systems. Phys. Rev. D
**11**, 809 (1975)CrossRefADSGoogle Scholar - 3.Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Academic Press, Boston (1994)Google Scholar
- 4.Cetto, A.M., de la Peña, L., Valdés-Hernández, A.: Quantization as an emergent phenomenon due to matter-zeropoint field interaction. J. Phys.
**361**, 012013 (2012)Google Scholar - 5.Cetto, A.M., de la Peña, L.: The Quantum Dice, an Introduction to Stochastic Electrodynamics. Kluwer, Dordrecht (1996)Google Scholar
- 6.Boyer, T.H.: Asymptotic retarded van der Waals forces derived from classical electrodynamics with classical electromagnetic zero-point radiation. Phys. Rev. A
**5**, 1799 (1972)CrossRefADSGoogle Scholar - 7.Dressel, J., Bliokh, K.Y., Nori, F.: Classical field approach to quantum weak measurements. Phys. Rev. Lett.
**112**, 110407 (2014)CrossRefADSGoogle Scholar - 8.Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Academic Press, Boston (1994)Google Scholar
- 9.P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics, pp. 51–54, 123–128, 487–488. Academic Press, Boston (1994)Google Scholar
- 10.Griffiths, D.J.: Introduction to Electrodynamics, 3rd edn. Prentice-Hall, Upper Saddle River (1999)Google Scholar
- 11.Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, 4th edn, p. 207. Pergamon Press, New York (1987). (Eq. 75.10)Google Scholar
- 12.Jackson, J.D.: Classical Electrodynamics, 3rd edn, p. 749. Wiley, New York (1998). (Eq. 16.10)Google Scholar
- 13.Huang, W., Batelaan, H.: Dynamics underlying the Gaussian distribution of the classical harmonic oscillator in zero-point radiation. J. Comput. Methods Phys.
**2013**, 308538 (2013)CrossRefGoogle Scholar - 14.Thornton, S.T., Marion, J.B.: Classical Dynamics of Particles and Systems, 5th edn, pp. 117–128. Brooks/Cole, Belmont (2004)Google Scholar
- 15.Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn, p. 80. Butterworth-Heinemann, Oxford (1976)Google Scholar
- 16.Yariv, A.: Quantum Electronics, 3rd edn, p. 407. Wiley, New York (1988)Google Scholar
- 17.Heisenberg, W.: Über quantentheoretische Umdeutung kinematischer und mechanischer Besiehungen. Z. Phys.
**33**, 879 (1925)CrossRefADSMATHGoogle Scholar - 18.Aitchison, I.J.R., MacManus, D.A., Snyder, T.M.: Understanding Heisenbergs magical paper of July 1925: a new look at the calculational details. Am. J. Phys.
**72**, 11 (2004)CrossRefMathSciNetGoogle Scholar - 19.Averbukh, V., Moiseyev, N.: Classical versus quantum harmonic-generation spectrum of a driven anharmonic oscillator in the high-frequency regime. Phys. Rev. A
**57**, 1345 (1998)CrossRefADSGoogle Scholar - 20.Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn, p. 366. Upper Saddle River, Pearson Prentice Hall (2005)Google Scholar