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American Journal of Physics

Dispersion relation for de Broglie waves

D. A. L. Paul

Published Apr 1, 1980• 5 citations
Publisher

Abstract

The dispersion relation for de Broglie waves is examined sufficiently to remove a possible ambiguity in the de Broglie frequency. The motion of a heavy (guided) photon, which obeys the same dispersion relation, is then studied in a gravitational field, leading to the Einsteinian relation frequency shift in a gravitational field.
DOI
10.1119/1.12140

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This paper is about a new paradigm shift in physics. According to Einstein, Newton’s theory of gravity is a first approximation of general relativity, which is Einstein’s theory of gravity, in the limit, which is when gravitational fields are weak and the velocities of masses (relative to a coordinate system) are small compared to the velocity of light. Since Schrödinger was aware of de Broglie’s relation when he formulated his quantum wave mechanics, it is acceptable to enforce the application of de Broglie’s relation in Schrödinger’s quantum wave mechanics. This enforcement leads to a new paradigm shift in physics.
The de Broglie wave phenomenon was studied in the framework of Lagrangean formulation of Relativistic Mechanics. The conclusion was made that the de Broglie wave of a particle is a relativistic phenomenon related to the field-dependent proper mass. The wave develops in a process of proper-to-kinetic mass transformation when the particle is given a momentum. The time-part of the 4-wave vector is related to the proper mass oscillation while the spacial part is due to the momentum; both parts can be treated in terms of virtual (longitudinal and scalar) photons characterizing excitation states of a mediating scalar field. It is shown that in Relativistic Mechanics with the variable proper mass both electromagnetic and gravitational force can be treated on the same footing: both are due to the same source that is, the proper mass. Consequently, the scalar mediating field provides a mechanism of unification of gravitational and electromagnetic forces. The concept of the unified divergence-free field is suggested, in which the real (transverse) photon represents a pure space-like particle while the neutrino is a pure time-like particle. In this connection, the parity problem is discussed.
Abstract Properties of de Broglie waves and their differences from any other waves, for example, electromagnetic, are discussed. Their little-known properties are given: 1) the propagation velocity is greater than the speed of light V > c ; 2) dispersion even in vacuum (waves of different frequencies propagate at different speeds, which results in the spreading of any wave packets associated with the particles); 3) the connection of the de Broglie wave with another physical wave moving with a speed v less than the speed of light, but so always v · V = c 2, and for the photons v = V = c. According to later views of de Broglie, this second wave is not the group wave. Possible mechanisms of generating the de Broglie waves based on spherical and cylindrical hollow quantum resonators, which can be specially created in matter using specific nanostructures are considered. Perhaps their nature itself realizes them for any micro-objects, regardless of their electrical, magnetic and gravitational properties. That is, this phenomenon is universal. Applications and realizations of this idea are discussed in the part II of our article.