Abstract The present paper concerns a formalism aimed to describe the behavior and trajectories of complex scalar waves with space-varying amplitude in media characterized by variable refractive index. The classical wave equation is transformed into an eikonal-type equation, without imposing the geometrical optics limit, satisfied by an effective refractive index. The effective index combines the refractive properties of the medium with self-refraction phenomena generated by the space dependence of the wave amplitude. It also induces wave packet dispersion, leading to relations between the group and phase velocities that are particularly relevant in the context of de Broglie waves. The equations of motion are provided, and it is shown that the self-refraction term bends the trajectory of the wave and changes its velocity. The dynamics is dictated by the curvature of the trajectory expressed in terms of the gradient of the effective refractive index. The uncovered findings point toward a possible generalization of geometrical optics with relevant implications for its relation to quantum mechanics, general relativity and transformation optics.
Abstract The present paper concerns a formalism aimed to describe the behavior and trajectories of complex scalar waves with space-varying amplitude in media characterized by variable refractive index. The classical wave equation is transformed into an eikonal-type equation, without imposing the geometrical optics limit, satisfied by an effective refractive index. The effective index combines the refractive properties of the medium with self-refraction phenomena generated by the space dependence of the wave amplitude. It also induces wave packet dispersion, leading to relations between the group and phase velocities that are particularly relevant in the context of de Broglie waves. The equations of motion are provided, and it is shown that the self-refraction term bends the trajectory of the wave and changes its velocity. The dynamics is dictated by the curvature of the trajectory expressed in terms of the gradient of the effective refractive index. The uncovered findings point toward a possible generalization of geometrical optics with relevant implications for its relation to quantum mechanics, general relativity and transformation optics.
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.
A small-wavelength, quasi-geometric optics is developed. The approach is based on an amplitude wave function that represents both the probability amplitude of the geometrical rays and the physical-optics disturbance. All ray paths contribute phases to the amplitude function proportional to their optical paths, measured in units of normalized wavelength. However, the dominant contributions are due to those rays that satisfy Fermat’s principle. This formulation leads to an integral equation for the wave-amplitude function with a kernel function that is a path integral. The integral equation reduces, in weakly focusing paraxial media, to a solution of a Schrödinger type of partial differential equation. The classical ray variables are operators with expectation values satisfying the geometric-optics equations. The second moments of the ray operators, with minimum-uncertainty initial wave packet, define second-order optical parameters. A special case of the quasi-geometric-optics integral equation is the quasi-optical laser-mode equation, allowing the interpretation of the stationary modes as probability amplitudes. For media with arbitrary inhomogeneous index of refraction, a matrix wave equation with a two-component wave function is derived. The matrix ray operators also satisfy a correspondence principle. A physical interpretation of the new wave function is presented.
According to Schrodinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrodinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave functionΨ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,∂t∂R2+∇⋅[R2m∇S−qA]=0,∂t∂S+2m(∇S−qA)2+qϕ+U+Q=0,whereQ=−2mℏ2R∇2R. The gauge-invariant kinetic propagation momentum is thereforeΠ=∇S−qA,and the associated local propagation velocity isv=mΠ. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term Q provides an additional modification of the propagation structure. The same equations can be written in a second-order form,mdtdv=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.
According to Schrödinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.
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.