Starting from the time-independent Schrodinger equation for elastic scattering a theory of wave-particle duality is developed which takes into account effects due to the quantum potential by means of a generalised ray equation. The duality between waves and particles is expressed in terms of two basic equations. Wave effects are described by a nonlinear amplitude equation which is coupled to a generalized ray equation that describes particle effects. The coupling between these two aspects of duality occurs by means of a function called the quantum expansion coefficient which describes the way in which a bundle of ray paths expand as the propagation distance increases. For systems in which the phase of the one-body wavefunction is a separable function of its coordinates, the authors derive simplified forms for the ray equations. As an example the authors show how the theory may be used to calculate tunnelling ray paths in planar electron channelling. The reflected rays in this case originate from beyond the classical turning points, thereby demonstrating the existence of a Goos-Hanchen effect in electron channeling.
Starting from the time-independent Schrodinger equation for elastic scattering a theory of wave-particle duality is developed which takes into account effects due to the quantum potential by means of a generalised ray equation. The duality between waves and particles is expressed in terms of two basic equations. Wave effects are described by a nonlinear amplitude equation which is coupled to a generalized ray equation that describes particle effects. The coupling between these two aspects of duality occurs by means of a function called the quantum expansion coefficient which describes the way in which a bundle of ray paths expand as the propagation distance increases. For systems in which the phase of the one-body wavefunction is a separable function of its coordinates, the authors derive simplified forms for the ray equations. As an example the authors show how the theory may be used to calculate tunnelling ray paths in planar electron channelling. The reflected rays in this case originate from beyond the classical turning points, thereby demonstrating the existence of a Goos-Hanchen effect in electron channeling.
Abstract Wave-particle duality is the fundamental phenomenon of particles and fields in quantum mechanics. In the past, the trajectory-like (particle-like) behaviour and wave-like behaviour has been considered separately. In this article, a superimposed model is derived to characterise wave-particle duality of classical particles. The superimposed model reflects an invariant mathematical structure (analogous variables and differential equations) from classical mechanics, classical field theories and quantum mechanics. Its analytical solution carries trajectory-like property (phase-independent) and wave-like property (phase-dependent) of particles that is consistent with to Schrodinger’s picture. Subsequently, the presented model is applied to model duality of classical waves in electromagnetism, acoustics and elasticity. The analysis implies the existence of quantum effects of classical waves at macroscopic scale. It predicts quantum picture on energy and momentum exchange between classical particles and waves. As seen in the model, wave-particle duality reflects inherent and indispensable characteristics of classical objects.
Both classical and wave-mechanical monochromatic waves may be treated in terms of exact ray-trajectories (encoded in the structure itself of Helmholtz-like equations) whose mutual coupling is the one and only cause of any diffraction and interference process. In the case of Wave Mechanics, de Broglie’s merging of Maupertuis’s and Fermat’s principles (see Section 3) provides, without resorting to the probability-based guidance-laws and flow-lines of the Bohmian theory, the simple law addressing particles along the Helmholtz rays of the relevant matter waves. The purpose of the present research was to derive the exact Hamiltonian ray-trajectory systems concerning, respectively, classical electromagnetic waves, non-relativistic matter waves and relativistic matter waves. We faced then, as a typical example, the numerical solution of non-relativistic wave-mechanical equation systems in a number of numerical applications, showing that each particle turns out to “dances a wave-mechanical dance” around its classical trajectory, to which it reduces when the ray-coupling is neglected. Our approach reaches the double goal of a clear insight into the mechanism of wave-particle duality and of a reasonably simple computability. We finally compared our exact dynamical approach, running as close as possible to Classical Mechanics, with the hydrodynamic Bohmian theory, based on fluid-like “guidance laws”.
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.
The phase-shift equations are derived in a form that is convenient for the analysis of both the radial Schr\"odinger equation and the problem of wave propagation in nonhomogeneous media. In the case of bound quantum states, the conservation condition is discussed for the sum of the classical action (or eikonal) and the phase. The expression for the reflection coefficient describes exactly two opposite limiting cases: the Fresnel reflection and the short-wavelength limit. In the case of wave propagation in symmetric media, both transmission and reflection coefficients are given as elementary functions of the phase shifts only.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
Bohr's response in 1925 to the failure of the Bohr–Kramers–Slater theory was, as we have seen, to regard wave–particle duality as having a formal, rather than realistic, significance; he saw it as a ‘limitation of our usual means of visualization’. That same year, however, wave–particle duality was generalised by Louis-Victor de Broglie to cover the theory of matter. Moreover, the advent of quantum mechanics (i.e. matrix mechanics and wave mechanics), rather than dispelling the duality problem, heightened it. How did Bohr respond to these new developments in the quantum theory? A matter of waves In the autumn of 1923 de Broglie put forward the extremely bold hypothesis that any particle whatever is associated with a wave whose wavelength λ and frequency ν are proportional to the particle's momentum and energy respectively: λ = h / p , v = E / h . De Broglie regarded his hypothesis as the basis of a new mechanics, a wave mechanics related to classical mechanics in the same way as wave optics is related to ray optics. On this view the trajectory of a particle corresponds to the ray of the wave associated with it (i.e. the normal to the equiphase surface of the wave), and the velocity of the particle corresponds to the group velocity of the wave. De Broglie suggested, moreover, that in certain circumstances particles should exhibit diffraction effects: a stream of electrons, for example, which passes through a slit in a screen should undergo diffraction.