The behavior of classical and quantum wave beams in stationary media is shown to be ruled by a "Wave Potential" function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the quantum case, the form of Bohm's "Quantum Potential", which is therefore not so much a "quantum" as a "wave" property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the quantum case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
The behavior of classical and quantum wave beams in stationary media is shown to be ruled by a "Wave Potential" function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the quantum case, the form of Bohm's "Quantum Potential", which is therefore not so much a "quantum" as a "wave" property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the quantum case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
The behavior of classical and beams in stationary media is shown to be ruled by a Wave Potential function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the case, the form of Bohm's Quantum Potential, which is therefore not so much a quantum as a wave property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
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.
With an apparent delay of over one century with respect to the development of standard Analytical Mechanics, but still in fully classical terms, the behavior of classical monochromatic wave beams in stationary media is shown to be ruled by a dispersive "Wave Potential" function, encoded in the structure of the Helmholtz equation. An exact, ray-based Hamiltonian description, revealing a strong ray coupling due to the Wave Potential, and reducing to the geometrical optics approximation when this function is neglected, is shown to hold even for typically wave-like phenomena such as diffraction and interference. Recalling, then, that the time-independent Schroedinger equation (associating the quantum motion of mono-energetic particles with stationary monochromatic matter waves) is itself a Helmholtz-like equation, the mathematical treatment holding in the classical case is extended, without resorting to statistical concepts, to the exact, trajectory-based, Hamiltonian quantum dynamics of point-like particles. The particle trajectories and motion laws turn out to be coupled, in this case, by a function strictly analogous to the Wave Potential and formally assuming the familiar form of Bohm's "Quantum Potential", which is therefore not so much a "quantum" as a "wave" property - in whose absence the quantum particle dynamics reduces to the classical one. The time-independent Schroedinger equation is argued to be not a trivial particular case of the time-dependent one, but the exact quantum dynamical ground on which Schroedinger's time-dependent statistical description (representing particles as travelling wave-packets) is based. It provides indeed the (exact) link between classical particle dynamics and Bohm's hydrodynamics.
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.
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 indeterministic character of physical laws is generally considered to be the most important consequence of quantum physics. A deterministic point of view, however, together with the possibility of well defined Hamiltonian trajectories, emerges as the most natural one from the analysis of the timeindependent Helmholtz-like equations encountered both in Classical Electromagnetism and in Wave Mechanics. In the former case the rays of a beam are described, for any set of boundary conditions, by an exact approach avoiding the geometrical optics approximation; in the latter case the same pattern of lines is shown to provide the trajectories of a quantum particle beam, and to correspond to a set of deterministic dynamical laws containing the classical ones as a simple limiting case. Paper available at: http://arxiv.org/ftp/arxiv/papers/0706/0706.3102.pdf