In physics, the zitterbewegung (German pronunciation: [ˈtsɪtɐ.bəˌveːɡʊŋ], from German zittern 'to tremble, jitter' and Bewegung 'motion') is the theoretical prediction of a rapid oscillatory motion of elementary particles that obey relativistic wave equations.
This prediction was first discussed by Gregory Breit in 1928[1][2] and later by Erwin Schrödinger in 1930[3][4] 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 an apparent fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of 2mc2/ℏ, or approximately 1.6×1021 radians per second.
This apparent oscillatory motion is often interpreted as an artifact of using the Dirac equation in a single particle description.
For the hydrogen atom, the zitterbewegung is related to the Darwin term, a small correction of the energy level of the s-orbitals.
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.
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 transformation, when the positive and negative components are decoupled.
Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.
[7] More recently, it has been noted that in the case of free particles it could just be an artifact of the simplified theory.
It doesn't appear in the correct second quantized theory, or rather, it is resolved by using Feynman propagators and doing QED.
Nevertheless, it is an interesting way to understand certain QED effects heuristically from the single particle picture.
(or zig and zag components), where each is the source term in the other's equation of motion, with a coupling constant proportional to the original particle's rest mass
, as The original massive Dirac particle can then be viewed as being composed of two massless components, each of which continually converts itself to the other.
Since the components are massless they move at the speed of light, and their spin is constrained to be about the direction of motion, but each has opposite helicity: and since the spin remains constant, the direction of the velocity reverses, leading to the characteristic zigzag or zitterbewegung motion.
Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence.
[14] An optical analogue of zitterbewegung was demonstrated in a quantum cellular automaton implemented with orbital angular momentum states of light[15] Other proposals for condensed-matter analogues include semiconductor nanostructures, graphene and topological insulators.