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.