A matter wave refers to the wave-like behavior exhibited by particles of matter, based on the principles of quantum mechanics. This concept stems from the de Broglie hypothesis, proposed by Louis de Broglie in 1924, which posits that all matter possesses wave-like properties. According to de Broglie's theory, every particle has an associated wavelength, known as the de Broglie wavelength, which is inversely proportional to its momentum.
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A matter wave is the quantum wave associated with a material particle. Its phase can interfere after coherent paths recombine.
In the position representation, a matter wave is described by a wavefunction: a complex number probability amplitude, with detection probabilities given by the Born rule. A localized wave packet combines different momenta; its envelope can move while its individual quantum phase crests move at another speed. Neither curve is a literal mechanical oscillation of material along a particle's path. The de Broglie phase harmony explains the historical route to ; the relativistic matter-wave velocities distinguish the two speeds. The Davisson–Germer experiment and the Thomson electron transmission diffraction experiment connect that spatial quantum phase to measured electron diffraction.
