# Zitterbewegung

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Zitterbewegung

Zitterbewegung (English: "trembling motion", from German) is a theoretical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation. 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 (at the speed of light) of the position of an electron around the median, with a circular frequency of $2 m c^2 / hbar ,!$, or approximately 1.6e|21 Hz.

The time-dependent Schrödinger equation :$H psi \left(mathbf\left\{x\right\},t\right) = i hbar frac\left\{partialpsi\right\}\left\{partial t\right\} \left(mathbf\left\{x\right\},t\right) ,!$

where $H ,!$ is the Dirac Hamiltonian for an electron in free space

:$H = left\left(alpha_0 mc^2 + sum_\left\{j = 1\right\}^3 alpha_j p_j , c ight\right) ,!$

implies that any operator Q obeys the equation

:$-i hbar frac\left\{partial Q\right\}\left\{partial t\right\} \left(t\right)= left \left[ H , Q ight\right] ,!;.$

In particular, the time-dependence of the position operator is given by

:$hbar frac\left\{partial x_k\right\}\left\{partial t\right\} \left(t\right)= ileft \left[ H , x_k ight\right] = alpha_k ,!;$

where $alpha_k equiv gamma_0 gamma_k$.

The above equation shows that the operator $alpha_k$ can be interpreted as the kth component of a "velocity operator."

The time-dependence of the velocity operator is given by

:$hbar frac\left\{partial alpha_k\right\}\left\{partial t\right\} \left(t\right)= ileft \left[ H , alpha_k ight\right] = 2 \left[i gamma_k m - sigma_\left\{kl\right\}p^l\right] = 2i \left[p_k-alpha_kH\right] ,!;$

where $sigma_\left\{kl\right\} equiv frac\left\{i\right\}\left\{2\right\} \left[gamma_k,gamma_l\right]$.

Now, because both $p_k$ and $H$ are time-independent, the above equation can easily be integrated twice tofind the explicit time-dependence of the position operator. First:

:$alpha_k \left(t\right) = alpha_k \left(0\right) e^\left\{-2 i H t / hbar\right\} + c p_k H^\left\{-1\right\}$

Then:

:$x_k\left(t\right) = x_k\left(0\right) + c^2 p_k H^\left\{-1\right\} t + \left\{1 over 2 \right\} i hbar c H^\left\{-1\right\} \left( alpha_k \left(0\right) - c p_k H^\left\{-1\right\} \right) \left( e^\left\{-2 i H t / hbar \right\} - 1 \right) ,!$

where $x_k\left(t\right) ,!$ is the position operator at time $t ,!$.

The resulting expression consists of an initial position, a motion proportional to time, and an unexpected oscillation term with an amplitude equal to the Compton wavelength. That oscillation term is the so-called "Zitterbewegung."

Interestingly, the "Zitterbewegung" term vanishes on taking expectation values for wave-packets that are made up entirely ofpositive- (or entirely of negative-) energy waves. This can be achieved by taking a [http://en.wikipedia.org/wiki/Foldy-Wouthuysen_transformation Foldy Wouthuysen transformation] . Thus, we arrive at the interpretation of the "Zitterbewegung" as being caused byinterference between positive- and negative-energy wave components.

ee also

* Casimir effect
* Lamb shift
* Stochastic electrodynamics: Zitterbewegung is explained as an interaction of a classical particle with the zero-point field.
* Barut-Zanghi theory, a theory of classical relativistic electrons with spin produced by Zitterbewegung, which produces a nonlinear Dirac-like equation.

References and notes

* E. Schrödinger, "Über die kräftefreie Bewegung in der relativistischen Quantenmechanik" ("On the free movement in relativistic quantum mechanics"), Berliner Ber., pp. 418-428 (1930); Zur Quantendynamik des Elektrons, Berliner Ber, pp. 63-72 (1931)

* A. Messiah, "Quantum Mechanics Volume II", Chapter XX, Section 37, pp. 950-952 (1962)

* [http://modelingnts.la.asu.edu/pdf/ZBW_I_QM.pdf The Zitterbewegung Interpretation of Quantum Mechanics] , an alternative explanation in addition to positive-negative energy states interference.
* [http://www.newscientist.com/channel/fundamentals/mg19526112.300-the-word-zitterbewebung.html Zitterbewegung in New Scientist]

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