Exact quasi optical theory of self diffraction, finite waist focusing and reflection of matter wave trajectories
Standard Quantum Mechanics (SQM) is the most widely accepted "route" starting from the common "main road" provided by de Broglie's and Schrodinger's discovery of matter waves and of their eigen functions and eigen values. Another "route" is offered by Bohm's Mechanics, able to reproduce the same results of SQM, while interpreting the streamlines of probability current density as a set of "quantum trajectories" We exploit here a further and somewhat different "route" based on the fact that the stationary Schrodinger equation (as well as any other Helmholtz like equation) may be treated, bypassing any kind of quasi optical approximation, in terms of exact trajectories, coupled and guided by a stationary, energy dependent function (which we call "Wave Potential" encoded in the structure itself of Helmholtz like equations and acting normally to the relevant particle momentum. The basic property of the Wave Potential is therefore to leave the particle energy unchanged: a property which cannot be shared (in spite of its formal analogy) by the time dependent "Quantum Potential" of the Bohmian theory, involving the entire spectrum of possible eigen energies. This property turns out to allow the numerical computation of the particle trajectories in stationary external force fields, and to show that all diffractive processes consist of energy preserving exchanges between the longitudinal and transverse momentum components of each single particle. We compute and discuss the trajectories of particles moving, under the guiding rule of the Wave Potential, in many different force fields, such as a constant field and the ones due to a potential barrier, a potential step and a focalizing potential, respectively. The very existence of these exact, point particle, trajectory based solutions appears to reserve the uncertainty relations to wave packet particle representations.