Matterwave

Matter wave, or de Broglie wave, refers to the wave-like nature of matter as described by quantum mechanics. The concept arises from de Broglie's hypothesis that every particle with momentum p has an associated wavelength λ = h/p, where h is Planck's constant. For non-relativistic particles, p = mv, so λ = h/(mv). The corresponding energy is E = p^2/(2m), and a free particle’s time evolution is described by a wave with angular frequency ω = E/ħ and wave number k = p/ħ.

In quantum mechanics, the state of a particle is described by a wavefunction ψ(r,t). A plane wave solution has ψ ~ exp[i(k·r − ωt)]. A localized particle is represented by a wave packet, a superposition of plane waves, whose group velocity equals the particle’s velocity v = p/m. The phase velocity is v_p = ω/k = E/p = v/2 in the non-relativistic case. The probability density of finding the particle is given by |ψ|^2.

Matter waves have been confirmed by numerous experiments, including electron diffraction and interference (the Davisson–Germer experiment), neutron interferometry, and atom interferometry. More recently, interference has been observed with increasingly large molecules, illustrating the persistence of quantum coherence at larger scales. Applications of matter-wave concepts include electron microscopy, precision measurements, and tests of quantum foundations, as well as developments in atom optics and Bose–Einstein condensates.