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A non-probabilistic insight into Wave Mechanics

The behavior of classical monochromatic waves in sta- tionary media is shown to be ruled by a novel, frequency-dependent function which we call Wave Potential, and which we show to be enco- ded in the structure of the Helmholtz equation. An exact, Hamiltonian, ray-based kinematical treatment, reducing to the usual eikonal approxi- mation in the absence of Wave Potential, shows that its presence in- duces a mutual, perpendicular ray-coupling, which is the one and only cause of wave-like phenomena such as diraction and interference. The "piloting" role of the Wave Potential, whose discovery does already constitute a striking novelty in the case of classical waves, turns out to play an even more important role in the quantum case. Recalling, indeed, that the time-independent Schrodinger equation (associating the motion of mono-energetic particles with stationary monochromatic matter waves) is itself a Helmholtz-like equation, the exact, ray-based treatment developed in the classical case is extended - without resorting to statistical concepts - to the exact, trajectory-based Hamiltonian dy- namics of mono-energetic point-like particles. Exact, classical-looking particle trajectories may be defined, contrary to common belief, and turn out to be perpendicularly coupled and piloted by an exact, energy- dependent Wave Potential, similar in the form, but not in the physical meaning, to the statistical, energy-independent "Quantum Potential" of Bohm's theory, which is aected, as is well known, by the prac- tical necessity of representing particles by means of statistical wave packets, moving along probability flux lines. This result, together with the connection shown to exist between Wave Potential and Uncertainty Principle, allows a novel, non-probabilistic interpretation of Wave Me- chanics, in the original spirit both of de Broglie and Schrodinger.