In 1924 Louis de Broglie published a breakthrough hypothesis: matter has wave properties. Building on Einsteins proposal that the photoelectric effect can be described using quantized energy transfers and by Einsteins separate proposal, from special relativity, that mass at rest is equivalent to energy via , de Broglie proposed that matter in motion appears to have an associated wave with wavelength  where  is the matter momentum from the motion.  Requiring his wavelength to encircle an atom, he explained quantization of Bohr's orbits. Simultaneously this showed that the wave behavior of light was essentially a quantum effect.
De Broglie expanded the Bohr model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron is observed only in situations that permit a standing wave around a nucleus. An example of a standing wave is a violin string, which is fixed at both ends and can be made to vibrate. The waves created by a stringed instrument appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is related to the length of the vibrating object and the boundary conditions. For example, because the violin string is fixed at both ends, it can carry standing waves of wavelengths , where l is the length and n is a positive integer.  De Broglie suggested that the allowed electron orbits were those for which the circumference of the orbit would be an integer number of wavelengths. The electrons wavelength, therefore, determines that only Bohr orbits of certain distances from the nucleus are possible. In turn, at any distance from the nucleus smaller than a certain value, it would be impossible to establish an orbit. The minimum possible distance from the nucleus is called the Bohr radius. De Broglies treatment of the Bohr atom was ultimately unsuccessful, but his hypothesis served as a starting point for Schrödinger's wave equation.
Matter behaving as a wave was first demonstrated experimentally for electrons: a beam of electrons can exhibit diffraction, just like a beam of light or a water wave. Three years after de Broglie published his hypothesis two different groups demonstrated electron diffraction.  At the University of Aberdeen, George Paget Thomson and Alexander Reid passed a beam of electrons through a thin celluloid film, then later metal films, and observed the predicted interference patterns. (Alexander Reid, who was Thomson's graduate student, performed the first experiments but he died soon after in a motorcycle accident and is rarely mentioned.) At Bell Labs, Clinton Joseph Davisson and Lester Halbert Germer reflected an electron beam from a nickel sample in their experiment, observing well-defined beams predicted by wave models returning form the crystal. De Broglie was awarded the Nobel Prize in Physics in 1929 for his hypothesis; Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.
Building on de Broglies approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics and the non-relativistic Schrödinger equation as an approximation of the generalised case of de Broglies theory. Schrödinger subsequently showed that the two approaches were equivalent. The first applications of quantum mechanics to physical systems were the algebraic determination of the hydrogen spectrum by Wolfgang Pauli and the treatment of diatomic molecules by Lucy Mensing.