The Helmholtz Wave Potential: a non-probabilistic insight into Wave Mechanics

Abstract: The behavior of classical monochromatic waves in stationary media is shown to be ruled by a novel, frequency-dependent function which we call Wave Potential, and which we show to be encoded in the structure of the Helmholtz equation. An exact, Hamiltonian, ray-based kinematical treatment, reducing to the usual eikonal approximation in the absence of Wave Potential, shows that its presence induces a mutual, perpendicular ray-coupling, which is the one and only cause of wave-like phenomena such as diffraction and interference. 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 Schroedinger 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 dynamics 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 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 affected, as is well known, by the practical 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 Mechanics.