In physics, the zitterbewegung ("jittery motion" in German, ) is the predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first discussed by Gregory Breit in 1928 and later by Erwin Schrödinger in 1930 as a result of analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of , or approximately radians per second. For the hydrogen atom, zitterbewegung can be invoked as a heuristic way to derive the Darwin term, a small correction of the energy level of the s-orbitals. Theory Free fermion The time-dependent Dirac equation is written as , where is the reduced Planck constant, is the wave function (bispinor) of a fermionic particle spin-½, and is the Dirac Hamiltonian of a free particle: , where is the mass of the particle, is the speed of light, is the momentum operator, and and are matrices related to the Gamma matrices , as and . In the Heisenberg picture, the time dependence of an arbitrary observable obeys the equation In particular, the time-dependence of the position operator is given by . where is the position operator at time . The above equation shows that the operator can be interpreted as the -th component of a "velocity operator". Note that this implies that , as if the "root mean square speed" in every direction of space is the speed of light. To add time-dependence to , one implements the Heisenberg picture, which says . The time-dependence of the velocity operator is given by , where Now, because both and are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator. First: , and finally . The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the reduced Compton wavelength. That oscillation term is the so-called zitterbewegung. Interpretation as an artifact In quantum mechanics, the zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. The standard relativistic velocity can be recovered by taking a Foldy–Wouthuysen