2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169038The 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.The scientific content is unchanged. Abstract, Introduction and Conclusions were improved. The new version is given in 7 pages and double column and includes 6 figuresQuantum Physicstext