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Latin American journal of physics education

Deduction of the De Broglie's relation λ=h/p from the classical electrodynamics

Nizar Hamdan, H. Aktaa

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

It is well known that De Broglie enables the use of the relativistic mechanics and –depending on a number of assumptions – to discover his relation λ=h/p, which led further to the creation of the theory of quantum mechanics (QM). However, after De Broglie's approach famous contradictions have appeared between De Broglie's theory and the Special Relativity Theory, also there were a number of suggestions in order to solve these contradictions. According to the suggested method in our papers it was remarked a serious part of the contradictions, and by following this method, we will present a new approach to derive the relation λ=h/p, starting from classical electrodynamics without any contradictions between Special Relativity Theory (SRT) and De Broglie's theory.

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It is well known that De Broglie enables the use of the relativistic mechanics and –depending on a number of assumptions – to discover his relation λ=h/p, which led further to the creation of the theory of quantum mechanics (QM). However, after De Broglie's approach famous contradictions have appeared between De Broglie's theory and the Special Relativity Theory, also there were a number of suggestions in order to solve these contradictions. According to the suggested method in our papers it was remarked a serious part of the contradictions, and by following this method, we will present a new approach to derive the relation λ=h/p, starting from classical electrodynamics without any contradictions between Special Relativity Theory (SRT) and De Broglie's theory.
We present a heuristic derivation of the Planck–Einstein relation E = hν and the de Broglie relation p = h/λ starting from a single hypothesis: the existence of an absolute minimum action h. From dimensional considerations alone, both relations follow immediately, with h identified a posteriori with Planck’s constant on the basis of experimental data. The work complements a previous paper in which special relativity is derived from a correspondence principle, suggesting a unified heuristic framework for the foundations of modern physics. Correspondence: gianroccoguerriero@libero.it or gwriterekphrasis@gmail.com
This paper provides an examination of the de Broglie relation, tracing its historical development from the quantum hypotheses proposed by Planck and Einstein to its covariant relativistic derivation. The discussion begins by situating de Broglie's seminal insight within the early framework of quantum theory. We then reconstruct his original heuristic derivation. The primary focus of this work, however, is the derivation of the de Broglie relation directly from the principles of special relativity, employing the four-momentum formalism. A comparative analysis between the heuristic and relativistic approaches underscores the universality and conceptual coherence of the latter. The paper concludes by highlighting the significance of relativistic mechanics in establishing a consistent foundation for wave-particle duality, further reinforcing this through a quantum field theoretical perspective.
Chronal dynamics predicts: p = ħ/(cτ) is the fundamental relation from which E = pc, λ = h/p and Lebedev's experiments follow. It is shown that the postulate of energy-dependent time τ(E) = ħ/E in combination with the classical expression for the momentum of a photon p = E/c uniquely leads to the relation p = ħ/(cτ). This relation is identical to de Broglie's formula for a photon (p = h/λ), if we take λ = cτ. Thus, the de Broglie wavelength is not a fundamental property of a particle, but a projection of its chronal quantum into space. offers a new interpretation of the pressure of light as the pressure of the flow of chronal rhythms.
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 paper the quantised version of Newton Second Law is derived assuming merely the existence of de Broglie matter-waves and their basic properties. At the same time we keep an eye towards interpretations of quantum mechanics and will realise that the two most different interpretations (Copenhagen interpretation and the de Broglie-Bohm theory) owe their difference to two fundamentally different approaches to `Harmonisation'. In this regard we shall see that the guiding equation of the de Broglie-Bohm theory currently found in literature is not the most complete equation possible; as a result we answer one of the important questions in interpreting quantum mechanics, namely that `when does the concept of classical path (trajectory) makes sense in quantum mechanics?' Moreover, in light of special-relativistic considerations we shall easily see that in the Number-Division approach, i.e. that of de Broglie-Bohm, the wave operator no longer appears, making it in turn impossible the application of Clifford algebras (Dirac's `square root' of the wave operator).