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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Inadequacies in De Broglie's Theory: rectifications, verifications, and applications

Himanshu Chauhan, Ravindra Kumar Sinha

Published Oct 1, 2013• 0 citations
Publisher

Abstract

The proposed paper calls attention towards the unobserved mathematical and conceptual inadequacies persisting in the wave-particle duality and matter wave’s concepts, given by Louis de Broglie. Matter wave’s frequency and phase velocity expressions, shown to be inappropriate, are the consequences of these inadequate concepts. The rectifications in these concepts are presented through the corrected implementation of analogy between light waves and matter waves and thus modified frequency and phase velocity expressions are introduced. The proposed expressions are free from all the inadequacies and negations, contrary to that confronted by de Broglie’s proposed expressions. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. A novel General Quantum Mechanical Wave Equation is proposed involving the modified phase velocity expression, which itself can precisely derive out Schrodinger’s and Dirac’s Equation.

DOI
10.1117/12.2023843

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The proposed paper calls attention towards the unobserved mathematical and conceptual inadequacies persisting in the wave-particle duality and matter wave’s concepts, given by Louis de Broglie. Matter wave’s frequency and phase velocity expressions, shown to be inappropriate, are the consequences of these inadequate concepts. The rectifications in these concepts are presented through the corrected implementation of analogy between light waves and matter waves and thus modified frequency and phase velocity expressions are introduced. The proposed expressions are free from all the inadequacies and negations, contrary to that confronted by de Broglie’s proposed expressions. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. A novel General Quantum Mechanical Wave Equation is proposed involving the modified phase velocity expression, which itself can precisely derive out Schrodinger’s and Dirac’s Equation.
The proposed paper calls attention towards the unobserved mathematical and conceptual inadequacies persisting in the wave-particle duality and matter wave’s concepts, given by Louis de Broglie. Matter wave’s frequency and phase velocity expressions, shown to be inappropriate, are the consequences of these inadequate concepts. The rectifications in these concepts are presented through the corrected implementation of analogy between light waves and matter waves and thus modified frequency and phase velocity expressions are introduced. The proposed expressions are free from all the inadequacies and negations, contrary to that confronted by de Broglie’s proposed expressions. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. A novel General Quantum Mechanical Wave Equation is proposed involving the modified phase velocity expression, which itself can precisely derive out Schrodinger’s and Dirac’s Equation.
The proposed paper presents the unobserved inadequacies in de Broglie's concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc2 is shown to be inappropriate for matter waves and is acceptable only for photons. The superluminal phase velocity expression c2/\u{psion}, for matter waves, is investigated in detail and is also reported to be inadequate. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications. A new relation between the Kinetic energy and Momentum of the moving body is proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.
In this paper, we have considered that the particle velocity in the matter wave concept of de Broglie is relative. It reveals that matter wave associated with a moving particle is a photon wave which has longitudinal and transverse values. The conventional de-Broglie’s wavelength-momentum relation corresponds to the longitudinal wave. As a consequence, we have found the longitudinal and transverse relative values of the mass, time, length and velocity including the missing equations of Einstein’s theory of special relativity. It also reveals that momentum and energy are relative quantities and the corresponding expressions for the longitudinal and transverse values for both of them have been presented. Moreover, the addition rules, with a general expression, for all the relative quantities based on Lorentz-Einstein transformation law for the addition of velocities have been given. Finally, the maximum and minimum values of the relative quantities have been expressed in terms of Plank’s constant and the speed of light. Their numerical values have also been calculated and presented in this paper.
Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.
In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.