Complete Hamiltonian description of wave-like features in classical and quantum physics

TL;DR

This paper derives an exact Hamiltonian framework from the Helmholtz and Schrödinger equations that describes wave-like phenomena such as diffraction and interference in both classical optics and quantum mechanics, extending beyond traditional approximations.

Contribution

It introduces a unified Hamiltonian system that captures wave-like features in classical and quantum physics, surpassing the limitations of geometrical optics and classical mechanics.

Findings

Derivation of an exact Hamiltonian system from Helmholtz and Schrödinger equations.

Application of the system to both optical and quantum wave phenomena.

Recovery of classical mechanics as a limiting case.

Abstract

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system of equations describing in terms of ray trajectories a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the ordinary geometrical optics approximation, which is contained as a simple limiting case. Due to the fact that the time independent Schroedinger equation is itself a Helmoltz-like equation, the same mathematical solutions holding for a classical optical beam turn out to apply to a quantum particle beam, leading to a complete system of Hamiltonian equations which provide a set of particle trajectories and motion laws containing as a limiting case the ones encountered in classical Mechanics.