From September 1923 to October 10/kloc-0, Louis Victor de Broglie published three papers on wave and quantum in Bulletin of French Academy of Sciences. The first paper, entitled "Radiation-Wave and Quantum", points out that physical particles also have wave-particle duality, and there is a sine wave corresponding to moving particles. They always keep the same phase. Later, he called this imaginary immaterial wave phase wave. He considered the relativistic effect of a moving particle with a rest mass of m0, and regarded the corresponding internal energy m0c2 as a simple periodic phenomenon with a frequency of v0. He applied the concept of phase wave to electrons moving around the nucleus in a closed orbit, and deduced Bohr quantization conditions. In the third paper entitled "Quantum Gas Motion Theory and Fermat Principle", he further pointed out: "Only when the phase wave resonance is satisfied, is the stable orbit." In the second year's doctoral thesis, he wrote more clearly: "The resonance condition is l=nλ, that is, the circumference of the electron orbit is an integer multiple of the phase wavelength."

In the second paper entitled "Optics-Optical Quantum, Diffraction and Interference", De Broglie put forward the following hypothesis: "Under certain conditions, any moving particle can be diffracted. Diffraction will occur when the electron group passes through a relatively small opening. It is in this respect that it is possible to find the experimental verification of our views. "

De Broglie did not explicitly put forward the concept of matter wave here. He just used the concept of phase wave or phase wave and thought that there might be a hypothetical immaterial wave. But what kind of waves are they? At the end of his doctoral thesis, he specifically stated: "I deliberately blurred the phase wave and the periodic phenomenon, just like the definition of light quantum, which can be said to be only an explanation, so it is best to regard this theory as a statement with unclear physical content, not a final theory." Material wave was put forward by Schrodinger when he interpreted the physical meaning of wave function after the establishment of Schrodinger equation. De Broglie did not explicitly put forward the relationship between wavelength λ and momentum p: λ=h/p(h is Planck constant), but later it was found that this relationship was implicit in his paper, so it was called De Broglie formula.

From the end of 1925 to the beginning of 1926, inspired by A. Einstein's quantum theory of monatomic ideal gas and L. V. De Broglie's material wave hypothesis, Schrodinger put forward the wave mechanics equation corresponding to wave optics from the analogy of classical mechanics and geometric optics, which laid the foundation of wave mechanics. He initially tried to establish the theory of relativity. The wave equation later called Klein-Gordon equation (see field equation) is obtained. However, because the spin of electrons was not known at that time, the theory about the spectral fine structure of hydrogen atoms was not consistent with the experimental data. Later, he used the non-relativistic wave equation later called Schrodinger equation to deal with electrons, and the results were consistent with the experimental data.