Zitterbewegung

The dithering is a theoretical, fast movement of elementary particles, especially electrons, which obey the ( relativistic ) Dirac equation.

The existence of such a movement was postulated in 1930 by Erwin Schrödinger, as a result of his analysis of wave packet solutions of the Dirac equation for relativistic electrons in a vacuum. This interference between the positive and negative energy state produces a fluctuation in the position of the electron around the average value with an angular frequency of

With

• The electron mass
• The speed of light
• The reduced Planck 's constant.

The Zitterbewegung of a free relativistic particle has never been observed, but the behavior of such a particle was simulated with a caged ion by placing of it in an environment so that the non - relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation has (although the physical situation is different).

Theory

From the time-dependent Schrödinger equation

Where the Dirac Hamiltonian for an electron in vacuum

And the wave function,

Follows in the Heisenberg picture, that every operator Q obeys the following equation:

Specifically, the time-dependent position operator is given by

With.

The above equation shows that the operator can be interpreted as a k- th component of the " speed of the operator ."

The time dependence of the velocity is given by the operator

And wherein the pulse.

Because both are independent of time, the above equation can be integrated twice to obtain the explicit time dependence of the spatial operator. First:

Then:

The resulting expression consists of

• An initial position
• A component of motion is proportional to the time and
• An unexpected oscillation component ( " dither " ) having an amplitude corresponding to the Compton wavelength.

Interestingly, the Zitterbewegung term vanishes if one takes the expectation values ​​for wave packets (in whole or in waves with negative energy ) are made entirely of waves with positive energy. This can be achieved by the Foldy - Wouthuysen transformation.