The Dynamics of Wave-Particle Duality

Abstract

Both classical and wave-mechanical monochromatic waves may be treated in terms of exact ray-trajectories (encoded in the structure itself of Helmholtz-like equations) whose mutual coupling is the one and only cause of any diffraction and interference process. In the case of Wave Mechanics, de Broglie's merging of Maupertuis's and Fermat's principles (see Section 3) provides, without resorting to the probability-based guidance-laws and flow-lines of the Bohmian theory, the simple law addressing particles along the Helmholtz rays of the relevant matter waves. The purpose of the present research was to derive the exact Hamiltonian ray-trajectory systems concerning, respectively, classical electromagnetic waves, non-relativistic matter waves and relativistic matter waves. We faced then, as a typical example, the numerical solution of non-relativistic wave-mechanical equation systems in a number of numerical applications, showing that each particle turns out to "dances a wave-mechanical dance" around its classical trajectory, to which it reduces when the ray-coupling is neglected. Our approach reaches the double goal of a clear insight into the mechanism of wave-particle duality and of a reasonably simple computability. We finally compared our exact dynamical approach, running as close as possible to Classical Mechanics, with the hydrodynamic Bohmian theory, based on fluid-like "guidance laws". Journal of Applied Mathematics and Physics redly leading to Bohm's Mechanics [2]-[7], with its probability-based guidance-laws and hydrodynamic flow-lines.

Trajectories and the perception of classical motion in the free propagation of wave packets

“…Also, the semi-classical limit of quantum mechanics is identical to the eikonal approximation in optics. The identification of particle classical trajectories in semi-classical wave mechanics is an exact parallel of the assignment of ray trajectories to light in wave optics; see, for example the work of Orefice et al 22,23 Unfortunately, the notations of optics and quantum mechanics differ so that the precise equivalence is not obvious. Here, the analogy is exposed in some detail, to equate physical parameters of the different notations and to illuminate further the concept of trajectories in quantum waves and rays in light waves.…”

Primary Assumptions and Guidance Laws in Wave Mechanics

“…In conclusion, any Helmholtz-like equation of the form (2) is associated to a stationary system of exact ray-trajectories: a basic information which was not available until 2009 [3]- [8].…”

“…The time-integration of the wave-mechanical systems AII-1 and/or AII-2 provides, in conclusion, de Broglie's missing link, without any further assumption and without resorting to any kind of probabilistic interpretation. The particle trajectories and time-tables are simply found, in fact [8], by assigning ( )…”

“…, V x z representing a lens-like focalizing structure [8]. The point-like focus (Figure 1) obtained in the eikonal limit, i.e.…”

“…We report here (from [8]) the Hamiltonian systems of particle trajectories holding, respectively, in the non-relativistic and in the relativistic case.…”