Whatever was going to replace classical physics in the description of radiation and atomic phenomena had to confront the difficult task of somehow reconciling the wave-like and particle-like aspects of light in a single structure. An important clue would come from Einstein’s E=mc2. We tend to want to associate mass (and linear momentum) with material particles. But the Planck–Einstein relation E=hν connects energy with frequency, a determinedly wave-like property. So, here are two very simple yet fundamental equations connecting energy to mass and energy to frequency. Can they be combined? Louis de Broglie thought so, and in 1923 he generalized the discovery made by Einstein in 1905 by extending it to all material particles, and most notably to electrons. The de Broglie relation λ=h/p can be derived quite straightforwardly by comparing the Lorentz transformations for energy and momentum with those for a system of plane waves.
This work offers an original view of the internal structure of Planck's constant h through the prism of classical Maxwellian electrodynamics and the complex geometry of phase space. This presents a development of the chronal field interpretation of quantum phenomena proposed earlier by the author. Based on the concept of the photon as a dynamic phase node of field switching (h=π+1), a revision of classical paradoxes of quantum mechanics is carried out. A generalized energy equation E=ℏ/(τ0+τtransition) is formulated, eliminating the duality of rest mass and photon momentum. Bohr's postulates and the structure of electronic orbits are explicated as topological invariants of phase resonance (ei2πn=1). A geometric model of the Zeeman effect is proposed as a dynamic asymmetry of counter-propagating complex rotations of the field under the influence of an external magnetic potential. This work lays the foundation for causal chronodynamics of the microcosm without the use of probabilistic Copenhagen description. This presents an alternative deterministic interpretation of quantum mechanics, eliminating the probabilistic Copenhagen postulate. The author introduces the concept of the chronal field τ(r,t) as the local time of one radian of phase. The wave function Ψ is explicated as a real physical deformation of this field.
The de Broglie relation serves as a foundational pillar connecting classical linear momentum with quantum mechanical wave properties. For massive particles moving at relativistic speeds, momentum is routinely expressed using the Lorentz factor, yielding a direct identity with the quantum wave parameters. However, when applied to massless particles such as photons traveling at the speed of light, this product encounters an indeterminate mathematical form. This paper demonstrates how the general relativistic energy momentum dispersion relation resolves this breakdown. Through two detailed numerical case studies involving a relativistic electron moving at sixty percent of light speed and a green photon, it shows that while the algebraic expression relying on rest mass fails for massless entities, the underlying physical equivalence between momentum and wavelength remains globally valid across all particle regimes.
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.
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.
The Compton wavelength is conventionally introduced as the rest-mass scale $\lc=h/(mc)$, whereas moving particles are normally assigned the de Broglie wavelength $\ldb=h/p$. Haug recently introduced and subsequently refined a frame-dependent relativistic Compton wavelength, $\lcr=h/(\gamma mc)$, with reduced form $\lbarcr=\hbar/(\gamma mc)$. For a massive free particle with
p=γmv, the exact identity $\ldb=(c/v)\lcr=\lcr/\beta$ follows immediately. This article examines the consequences of taking $\lcr$, rather than $\ldb$, as the primary wavelength parameter for single-particle relativistic wave mechanics. The distinction is conceptually sharp at rest: $\lcr$ is finite and equals $\lc$ at
v=0, while
h/p is not defined by ordinary division at
p=0, although its limiting spatial wavelength is infinite and its wave number
k=p/ℏ is well defined and vanishes. We show that free-particle phase relations, Schr\"odinger eigenstates, box quantisation, diffraction conditions, uncertainty relations, and the plane-wave sectors of the Klein--Gordon and Dirac equations can all be written using $\lbarcr$ and
β. In this parameterisation, $E=\hbar c/\lbarcr$ and $p=\hbar\beta/\lbarcr$, so the de Broglie wavelength is a derived spatial period generated by the velocity-dependent spatial gradient of a Compton-scale spacetime phase. The reformulation is algebraically equivalent to standard momentum-space quantum mechanics, but it exposes an unequal domain and an overlooked organisation of four-momentum. The quantity
E/c=γmc, already present as the temporal component of four-momentum, is precisely Haug's Compton momentum
pC=h/λC,r. Every finite de Broglie wavelength of a massive on-shell particle can be represented by the relativistic Compton wavelength and velocity through $\ldb=\lcr/\beta$.