A consistent description of the motion of a particle by wave propagation implies not only a geometrical equivalence between a trajectory and a ray but also a kinematical equivalence between the time rates at which the particle or wave is traversing its path. This may be tentatively achieved by using a wave configuration, known as a wave packet, represented as a continuous superposition of monochromatic waves. If the wave packet can be regarded as a point, having zero amplitude everywhere except in a very small region, its motion along a trajectory might be interpreted as the motion of a single particle. The representation of a particle in space and time by means of a highly localized wave packet is a basic assumption of wave mechanics. The behavior of the wave packet as a whole will account for experiments where particle-like properties are observed, whereas individual wavelength components of packet will appear in experiments which depend upon interference and diffraction effects.
A consistent description of the motion of a particle by wave propagation implies not only a geometrical equivalence between a trajectory and a ray but also a kinematical equivalence between the time rates at which the particle or wave is traversing its path. This may be tentatively achieved by using a wave configuration, known as a wave packet, represented as a continuous superposition of monochromatic waves. If the wave packet can be regarded as a point, having zero amplitude everywhere except in a very small region, its motion along a trajectory might be interpreted as the motion of a single particle. The representation of a particle in space and time by means of a highly localized wave packet is a basic assumption of wave mechanics. The behavior of the wave packet as a whole will account for experiments where particle-like properties are observed, whereas individual wavelength components of packet will appear in experiments which depend upon interference and diffraction effects.
The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. Wave packet is often taken as the best approximation of quantum theory to a classical particle. If the wave packet is regarded as a point, with zero amplitude everywhere except in a very small region, then its motion can be interpreted as the motion of a single particle. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron wave packets. The localized wave packet solution disperses or spreads in time, however, at a later time, it reforms and appears as its initial state. Revivals in quantum systems were first studied in the Jaynes–Cumming model, describing a two-level atom interacting with a resonant monochromatic field.
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.
An attempt to explain the motion of particle is made in terms of a wave superposition. There is no localized superposition of waves that is supposed to be a single particle apart from other wave superposition, but there are a bunch of wave packets, which appears to be particle within a wave superposition that is made of individual waves around the space. There are countless number of waves that become a wave superposition, and the position of each wave packet is determined by the relative distance between packets, which is a superposition state. When a wave superposition interfere with a individual wave, this distance changes, and the position of each wave packet is fixed , which is called a measurement. With respect to the position of a particle, thus the particle is placed at the different position simultaneously in the superposition state. A change of the relative distance between wave packets appears to be a motion of particle, which is interpreted as a force acting on a particle. The photo-electric effect is explained in the same way, by which a superposition of charge waves interferes with the light wave, which increases the relative distance of wave packets in the atom, and it brings the outermost wave packet apart from the atom. The resistance in conductor is also understood by the wave interference with a superposition of charge waves residing in the conductor.
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.
A physics-oriented review is first given of wave equations: examples, the Cauchy problem, solutions in various coordinates, their symmetries, how to build a wave packet. Fourier analysis and dispersion relations are then introduced, and here the key tool is the symbol of differential operators, which makes it possible to study them in terms of algebraic polynomials involving cotangent-bundle variables. Further basic material deals with geometrical optics from the wave equation, phase and group velocity (and their dual relationship in momentum space), the Helmholtz equation and the eikonal approximation for the scalar wave equation. The Schrödinger equation is then derived with emphasis on the wave packet and its relation to classical behaviour in the light of the Einstein–de Broglie relation. For this purpose, it is shown that it is possible to build a wave packet for which the wave-like properties manifest themselves for distances of the order of atomic dimensions. The Fourier transform with respect to time of the wave packet obeys the stationary Schrödinger equation, while the wave function is found to obey a partial differential equation which is of first order in time. The wave equation The wave equation occurs in several branches of classical physics, e.g. the theory of sound, electromagnetic phenomena in vacuum and in material media, and elastic vibrations of material bodies. We consider first the simplest situation, i.e. the propagation of waves in a homogeneous, isotropic and stationary medium. Propagation is characterized by the refractive index n , a quantity that is independent of the point in space-time.