Beyond quasi-optics: an exact approach to self-diffraction, reflection and finite-waist focusing of matter wave trajectories

The "main road" open by de Broglie's and Schroedinger's discovery of matter waves and of their eigen-functions branched off, as is well known, into different "sub-routes". The most widely accepted one is Standard Quantum Mechanics (SQM), interpreting the time-dependent Schroedinger equation as the basic evolution law of a wave-packet which represents the simultaneous probabilistic permanence of a particle in its full set of eigenstates. Another "sub-route" is offered by Bohm's Mechanics, able to reproduce the same results of SQM, while interpreting the stream-lines of the probability current density as the "quantum trajectories" of the moving particles. Reminding that the so-called quasi-optical approximation represents a standard mathematical technique allowing a ray-based treatment of wave-like features, we present here an exact wave-mechanical "sub-route", based on the observation that the time-independent Schroedinger equation may be treated, bypassing any quasi-optical approximation, in terms of a Hamiltonian set of rays mutually coupled by an energy-dependent function (which we call "Wave Potential") encoded in the very structure of any Helmholtz-like equation. These rays lend themselves to be interpreted as the exact wave-dynamical trajectories and motion laws of classical-looking point-particles associated with the de Broglie-Schroedinger matter waves. The role of the Wave Potential, acting perpendicularly to the momentum of the moving particles, is to "pilot" them without any energy exchange: a property which isn't shared by the well-known "Quantum Potential" of the Bohmian theory, involving the entire spectrum of possible eigen-energies of a wave-packet. This property turns out to allow the numerical computation of the particle trajectories, which we perform and discuss here for particles piloted by the Wave Potential in many different and significant force-fields.