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.