Helmholtz wave trajectories in classical and quantum physics

TL;DR

This paper demonstrates that classical and quantum wave behaviors can be described by a unified Hamiltonian framework involving a 'Wave Potential', which accounts for diffraction, interference, and quantum effects without wave-packets.

Contribution

It introduces a deterministic Hamiltonian approach based on a 'Wave Potential' that unifies classical and quantum wave trajectories without relying on wave-packets.

Findings

Derives exact Hamiltonian trajectories for wave beams.

Shows the 'Quantum Potential' is a wave property, not exclusively quantum.

Reduces to classical optics and dynamics when coupling is neglected.

Abstract

The behavior of classical and quantum wave beams in stationary media is shown to be ruled by a "Wave Potential" function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the quantum case, the form of Bohm's "Quantum Potential", which is therefore not so much a "quantum" as a "wave" property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the quantum case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.