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Concept
Zitterbewegung
Summary
In physics, the zitterbewegung (ˈtsɪtɐ.bəˌveːɡʊŋ, ) 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 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 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 1.6e21 radians per second.
This apparent oscillatory motion is often interpreted as an artifact of using the Dirac equation in a single particle description and disappears when using quantum field theory. For the hydrogen atom, the zitterbewegung is related to the Darwin term, a small correction of the energy level of the s-orbitals.
The time-dependent Dirac equation is written as
where is the reduced Planck constant, is the wave function (bispinor) of a fermionic particle spin-1⁄2, and H 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 Q obeys the equation
In particular, the time-dependence of the position operator is given by
where xk(t) is the position operator at time t.
The above equation shows that the operator αk can be interpreted as the k-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 αk, one implements the Heisenberg picture, which says
The time-dependence of the velocity operator is given by
where
Now, because both pk and H are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.
Official source
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