Beyond the Eikonal Approximation in Classical Optics and Quantum Physics

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system
describing in terms of ray trajectories a very wide family of wave-like phenomena (including
diffraction and interference) containing the eikonal approximation as a simple limiting
case. Since, moreover, the time independent Schr¨odinger equation is itself a Helmholtz-like
equation, the same mathematics holding for a classical optical beam turns out to apply to
a quantum particle beam moving in a stationary force field, and the so called “quantum
potential” of the Bohmian theory turns out to be a particular case of a general wave-like
property. A suitable set of Hamiltonian equations is shown to provide both the exact and
deterministic trajectories of the particles and their dynamical laws, reducing to the laws of
classical mechanics in the eikonal limit. Wave-like features in Hamiltonian form are presented
in some typical numerical cases.