Editing Louis de Broglie (section)

=== Matter and wave–particle duality ===
{{Main|De Broglie hypothesis}}
Studying the nature of X-ray radiation and discussing its properties with his brother Maurice, who considered these rays to be some kind of combination of waves and particles, contributed to Louis de Broglie's awareness of the need to build a theory linking particle and wave representations. In addition, he was familiar with the works (1919–1922) of [[Marcel Brillouin]], which proposed a hydrodynamic model of an atom and attempted to relate it to the results of Bohr's theory. The starting point in the work of Louis de Broglie was the idea of Einstein about the [[Photon|quanta of light]]. In his first article on this subject, published in 1922, the French scientist considered blackbody radiation as a gas of light quanta and, using classical statistical mechanics, derived the [[Wien approximation|Wien radiation law]] in the framework of such a representation. In his next publication, he tried to reconcile the concept of light quanta with the phenomena of interference and diffraction and came to the conclusion that it was necessary to associate a certain periodicity with quanta. In this case, light quanta were interpreted by him as relativistic particles of very small mass.<ref name="Mehra">{{cite book |author = J. Mehra. |editor= J. Mehra. |title= The Golden Age of Theoretical Physics| chapter=Louis de Broglie and the phase waves associated with matter |edition=  |year= 2001 |publisher= World Scientific |pages = 546–570 }}</ref>

It remained to extend the wave considerations to any massive particles, and in the summer of 1923 a decisive breakthrough occurred. De Broglie outlined his ideas in a short note "Waves and quanta" ({{langx|fr|Ondes et quanta}}, presented at a meeting of the Paris Academy of Sciences on September 10, 1923),<ref>{{Cite web |date=1923 |title=Membres de l'Académie des sciences depuis sa création: Louis de Broglie Ondes et quanta |language=fr |url=https://www.academie-sciences.fr/pdf/dossiers/Broglie/Broglie_pdf/CR1923_p507.pdf |website=academie-sciences.fr}}</ref> which marked the beginning of the creation of wave mechanics. In this paper and his subsequent PhD thesis,<ref name="De_Broglie_PhD_English"></ref> the scientist suggested that a moving particle with energy ''E'' and velocity '''v''' is characterized by some internal periodic process with a frequency <math>E/h</math> (later known as [[Compton frequency]]), where <math>h</math> is the [[Planck constant]]. To reconcile these considerations, based on the quantum principle, with the ideas of special relativity, de Broglie associated wave he called a "phase wave" with a moving body, which propagates with the [[phase velocity]] <math>c^2/v</math>. Such a wave, which later received the name [[matter wave]], or [[de Broglie wave]], in the process of body movement remains in phase with the internal periodic process. Having then examined the motion of an electron in a closed orbit, the scientist showed that the requirement for phase matching directly leads to the quantum [[Bohr-Sommerfeld quantization|Bohr-Sommerfeld condition]], that is, to quantize the angular momentum. In the next two notes (reported at the meetings on September 24 and October 8, respectively), de Broglie came to the conclusion that the particle velocity is equal to the [[group velocity]] of phase waves, and the particle moves along the normal to surfaces of equal phase. In the general case, the trajectory of a particle can be determined using [[Fermat's principle]] (for waves) or the [[principle of least action]] (for particles), which indicates a connection between geometric optics and classical mechanics.<ref>[[Max Jammer]] ''The Conceptual Development of Quantum Mechanics''. New York: McGraw-Hill, 1966 2nd ed: New York: American Institute of Physics, 1989. {{ISBN|0-88318-617-9}}. Olivier Darrigol, "Strangeness and soundness in Louis de Broglie's early works", ''Physis'',  30 (1993): 303–372.</ref> The ''de Broglie wavelength'' {{math|''λ''}} is the [[Planck constant]] {{math|''h''}} divided by  [[momentum]] {{math|''p''}}:
<math display="block"> \lambda = \frac{h}{p}.</math>

This theory set the basis of wave mechanics. It was supported by Einstein, confirmed by the [[Davisson–Germer experiment|electron diffraction experiments]] of [[George Paget Thomson]] in the United Kingdom and [[Clinton Davisson]] and [[Lester Germer]] in the United States, and generalized by the work of Erwin Schrödinger.

Originally, de Broglie thought that real wave (i.e., having a direct physical interpretation) was associated with particles. However, when the wave aspect of matter was formalized by a [[wavefunction]] defined by the [[Schrödinger equation]], it came out as a pure mathematical entity having a probabilistic interpretation, without the support of physical elements. This wavefunction gives wave behavior to matter but it is only observed through individual quantum samples. However, in 1956 de Broglie again attempted a theory of a direct and physical interpretation of matter-waves, following the work of [[David Bohm]] and suggestions of [[Jean-Pierre Vigier]].<ref name=Bridgman-1960>{{Cite journal |last=Bridgman |first=P. W. |last2=de Broglie |first2=Louis |last3=Knodel |first3=Arthur J. |last4=Miller |first4=Jack C. |date=1960 |title=Review of Non-Linear Wave Mechanics: A Causal Interpretation, de BroglieLouis, KnodelArthur J., MillerJack C. |url=https://www.jstor.org/stable/24940668 |journal=Scientific American |volume=203 |issue=4 |pages=201–206 |issn=0036-8733}}</ref>