Zitterbewegung Zitterbewegung is a predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations . The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his 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 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 for a free fermion The time-dependent Dirac equation is written as where is the Planck constant , is the wave function 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". 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 Compton wavelength . That oscillation term is the so-called zitterbewegung. The zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- energy waves. This can be achieved by taking a Foldy–Wouthuysen transformation . Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.Experimental simulation Zitterbewegung of a free relativistic particle has never been observed directly, although there is strong evidence in favor of its existence. It has also been simulated twice in model systems that provide condensed-matter analogues of the relativistic phenomenon. The first example, in 2010, placed a trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation. Then, in 2013, it was simulated in a setup with Bose–Einstein condensates . Other proposals for condensed-matter analogues include graphene and topological insulators .
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