# Martins, Roberto de Andrade: *Louis de Broglie's Struggle with the Wave-Particle Dualism, 1923-1925*

Although the main lines of Louis de Broglie’s attempt to build a wave-particle theory are well known, the development of his views has not received a detailed analysis. This paper will focus upon the conflicts and hesitations that occurred in his work, from 1923 (his first papers on wave-particle duality) to 1925 (before the development of quantum mechanics proper). Starting from the relations E=mc² and E=hn, de Broglie attempted to associate a pulsating phenomenon to electrons and other particles, but at first he could not obtain suitable relativistic equations. He found a solution to this difficulty by introducing a wave, instead of a beat. The meaning of this wave was not clear, and its speed was greater than c, therefore introducing a new conflict with relativity. De Broglie introduced the idea of wave groups to solve this problem, and proved that the group velocity (and energy velocity) was equal to the velocity of the electron. The wave group also allowed de Broglie to ascribe a localization to the electron. However, this introduced a new conceptual problem. In a classical framework, to each electron should be assigned a well-defined speed, and therefore a well-defined energy and frequency. It was difficult to understand what meaning could be ascribed to a wave group, with a set of similar but different frequencies. This problem occurred four years before the proposal of Heisenberg’s uncertainty principle, and de Broglie did not arrive to similar ideas, at that time. In his thesis, de Broglie proposed a new approach to the problem, describing the electron itself as an extended system, since its electromagnetic energy is not concentrated in a point, but is spread over the whole space around the charge (with a stronger concentration around a centre). This concept is not equivalent to Schrödinger’s later proposal of an electron with extended charge, and it was not altogether clear in de Broglie’s thesis whether the charge itself was localized or spread around a centre. In the rest frame of the electron, its whole (infinite) structure was supposed to be pulsating with a frequency given by hn0=m0c². Applying the Lorentz transformation to this pulsation, de Broglie readily showed that the beat would transform to a wave, relative to other reference frames, and obtained the speed, frequency and other properties of the wave. In this new approach, since the electron has a strong energy concentration around a centre, the wave associated to its pulsation will also have a strong amplitude concentration around a centre traveling with the speed of the electron. This is mathematically equivalent to a wave group, but conceptually it is quite different, because in the rest frame it does have a single, well-defined frequency, keeping the classical (non-probabilistic) outlook that guided de Broglie’s work. The strong link between wave and extended particle, as presented in the thesis, did not solve other problems, however. De Broglie was attempting to develop a theory which as intended to describe both electromagnetic radiation (light quanta) and electrons and other particles. He envisaged light quanta as energy chunks with speed smaller than c (but very close to c) associated to electromagnetic waves with speed slightly greater than c (of course, this required a correction of Maxwell’s equations). To explain interference phenomena – both in light experiments and with electrons – he introduced a probabilistic relation between waves and energy quanta. The probability of absorption or emission of a light quantum (or electron) should be proportional to the intensity (square of the amplitude) of the wave at each region. This hypothesis was necessary to obtain an agreement between his theory and the classical optical results, but could not be derived from (and was hardly compatible with) his fundamental concept of wave-particle duality. In the papers he published before the thesis, in his thesis, and in papers published shortly after it, de Broglie was fighting against severe conceptual difficulties such as those. He kept changing some of his fundamental hypotheses, maintaining only a few basic assumptions, such as relativistic dynamics and the relation E=hn. Instead of a coherent and final theory, his papers exhibited a changeable work in progress, with deep and unsolved conceptual problems.