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. 1. Introduction 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 [1]. The first experimental evidence of the vacuum field dates back to 1947 when Lamb and his student Retherford found an unexpected shift in the hydrogen fine structure spectrum [2, 3]. The physical existence of the vacuum field has inspired an interesting modification to the classical mechanics, known as stochastic electrodynamics (SED) [4]. As a variation of classical electrodynamics, SED adds a background electromagnetic vacuum field to the classical mechanics. The vacuum field as formulated in SED has no adjustable parameters except that each field mode has a random initial phase and the field strength is set by the Planck constant, . With the aid of this background field, SED is able to reproduce a number of results that were originally thought to be pure quantum effects [1, 4–8]. 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 of an SED harmonic oscillator are identical to those of the ground state quantum harmonic oscillator [9]. As a consequence, the Heisenberg minimum uncertainty relation is satisfied, and the probability distributions of an SED harmonic oscillator are a Gaussian, identical to that of the ground state quantum harmonic oscillator. While in classical mechanics it is most
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