Standard 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.
Standard 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.
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 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.
The behavior of monochromatic electromagnetic waves in stationary media is shown to be ruled by a frequency dependent function, which we call Wave Potential, encoded in the structure of the Helmholtz equation. Contrary to the common belief that the very concept of "ray trajectory" is reserved to the eikonal approximation, a general and exact ray-based Hamiltonian treatment, reducing to the 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 any typically wave-like phenomenon, such as diffraction and interference. Recalling, then, that the time-independent Schroedinger and Klein-Gordon equations (associating stationary "matter waves" to mono-energetic particles) are themselves Helmholtz-like equations, the exact, ray-based treatment developed for classical electromagnetic waves is extended - without resorting to statistical concepts - to the exact, trajectory-based Hamiltonian dynamics of mono-energetic point-like particles, both in the non-relativistic and in the relativistic case. The trajectories turn out to be perpendicularly coupled, once more, by an exact, stationary, energy-dependent Wave Potential, coinciding in the form, but not in the physical meaning, with the statistical, time-varying, energy-independent "Quantum Potential" of Bohm's theory, which views particles, just like the standard Copenhagen interpretation, as traveling wave-packets. These results, together with the connection which is shown to exist between Wave Potential and Uncertainty Principle, suggest a novel, non-probabilistic interpretation of Wave Mechanics.
The behavior of monochromatic electromagnetic in stationary media is shown to be ruled by a frequency dependent function, which we call Wave Potential, encoded in the structure of the Helmholtz equation. Contrary to the common belief that the very concept of ray trajectory is reserved to the eikonal approximation, a general and exact ray-based Hamiltonian treatment, reducing to the 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 any typically wave-like phenomenon, such as diffraction and interference. Recalling, then, that the time-independent Schroedinger and Klein-Gordon equations (associating stationary matter waves to mono-energetic particles) are themselves Helmholtz-like equations, the exact, ray-based treatment developed for classical electromagnetic is extended - without resorting to statistical concepts - to the exact, trajectory-based Hamiltonian dynamics of mono-energetic point-like particles, both in the non-relativistic and in the relativistic case. The trajectories turn out to be perpendicularly coupled, once more, by an exact, stationary, energy-dependent Wave Potential, coinciding in the form, but not in the physical meaning, with the statistical, time-varying, energy-independent Quantum of Bohm's theory, which views particles, just like the standard Copenhagen interpretation, as traveling wave-packets. These results, together with the connection which is shown to exist between Wave Potential and Uncertainty Principle, suggest a novel, non-probabilistic interpretation of Wave Mechanics.
We present here the exact, trajectory based wave-mechanical description of particle dynamics provided by the time-independent Schroedinger equation, of which the time-dependent equation allows a statistical treatment. Let us remind, for comparison, that Bohmian mechanics is largely recognized to provide a basically hydrodynamic formulation of the quantum theory, describing the evolution of physical systems in terms of statistical streamlines which indicate the regions of space where the moving particles preferentially gather: the very concept of true particle path is avoided, in Bohmian theory, by a weak, non-perturbing measurement. In the present paper the wave-dynamical Hamiltonian equations providing the exact particle trajectories are directly obtained from the time-independent Schroedinger equation, and numerical solutions are presented for a number of different external potentials acting on the moving particles. A Wave-Potential function (holding for any - even classical - wave process described by a Helmholtz-like equation) is shown to be the one and only cause of wave-like features such as diffraction and interference, and to act, for matter waves, without any wave-particle energy exchange.
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 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.
The time-independent Schroedinger and Klein-Gordon equations - as well as any other Helmholtz-like equation - were recently shown to be associated with exact sets of ray-trajectories (coupled by a "Wave Potential" function encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory.