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Heuristic Derivation of the Planck–Einstein and de Broglie Relations from the Concept of Minimum Action

Gianrocco Guerriero

Published May 9, 2026• 0 citations
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Abstract

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

DOI
10.5281/zenodo.20097910

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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.
This pedagogical monograph describes some of the fundamental views of the laws of physics. The derivations are, however, obtained from a rather unconventional point of view. The Lorentz transformations and the special theory of relativity are derived without mentioning the phenomenon of light, and the de Broglie relations in the wave-corpuscle parallelism are derived without the help of Planck's constant. By the use of Schrodinger's idea of quantization as an eigenvalue problem, the foundations of wave mechanics are discussed as a mathematical problem without reference to Planck's constant. Finally, the Kepler problem in the special theory of relativity is studied starting from the energy law, and the applications to the Hulse-Taylor binary pulsar indicate that more data about the unseen companion are needed before the interpretation of the present data may be taken as the ultimate proof of the validity of the general theory of relativity.
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.
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.
A brief historical review of the Principle of Least Action is given, and it's methodological importance is emphasized: the basic equations of Classical Mechanics can be derived from the conjunction of the Principle Least Action with the Galilean Principle of Relativity and the hypothesis of homogeneity and isotropy of space and the homogeneity of time. In classical electrodinamics, the situation is similar: one can build the action keeping now in mind the Einstein's Principle of Relativity. By means of the solution of the Euler-Lagrange equations one obtain the equations of motion of charges in given fields (Lorentz force) and the first pair of Maxwell equations. Also, starting from the sources of the field (charges and currents), the second pair of Maxwell equations is obtained.\nIn the case of the General Relativity, the trajectories of a free particle are given by the geodesic lines in space-time. The Einstein equations are also obtained by finding the extreme of some action. A brief historical review of quantum theory is given, starting from the introduction of the quantum of action in physics, to the Schrödinger equation, which can be obtained from a variational principle, and the Heisenberg Uncertainty Principle. Interesting is the role of the classical action in the formulation of quantum mechanics by means of the Feynman path integrals. In it, the classical trajectory in the motion of a system between two configurations, it corresponds the limit in which the Planck constant tends to zero. In quantum electrodinamics, it is required also to start from a Lagrangian together with the properties of space-time simmetry, the gauge invariance and the CPT invariance. Finally, mention is made to the electroweak theory and to quantum cromodynamics