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" (, presented at a meeting of the Paris Academy of Sciences on September 10, 1923), which marked the beginning of the creation of wave mechanics. In this paper and his subsequent PhD thesis, the scientist suggested that a moving particle with energy E and velocity v is characterized by some internal periodic process with a frequency (later known as Compton frequency), where 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 . 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 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.Max Jammer The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966 2nd ed: New York: American Institute of Physics, 1989. . Olivier Darrigol, "Strangeness and soundness in Louis de Broglie's early works", Physis, 30 (1993): 303–372.
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" (, presented at a meeting of the Paris Academy of Sciences on September 10, 1923), which marked the beginning of the creation of wave mechanics. In this paper and his subsequent PhD thesis, the scientist suggested that a moving particle with energy E and velocity v is characterized by some internal periodic process with a frequency (later known as Compton frequency), where 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 . 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 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 veloci!ty 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.Max Jammer The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966 2nd ed: New York: American Institute of Physics, 1989. . Olivier Darrigol, "Strangeness and soundness in Louis de Broglie's early works", Physis, 30 (1993): 303–372.