TITLE:
The Dynamics of Wave-Particle Duality
AUTHORS:
Adriano Orefice, Raffaele Giovanelli, Domenico Ditto
KEYWORDS:
Helmholtz Equation, Wave Potential, Hamilton-Jacobi Equation, Wave Mechanics, De Broglie’s Duality, Matter Waves, Guidance Laws, Schrödinger Equations, Klein-Gordon Equation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.9,
September
18,
2018
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”.