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 1.6×1021 Hz.

Zitterbewegung of a free relativistic particle has never been observed, but the behavior of such a particle has been simulated with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different). [1] [2]

Contents

Theory

The time-dependent Schrödinger equation

H \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t} (\mathbf{x},t) \,\!

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

H = \left(\alpha_0 mc^2 %2B \sum_{j = 1}^3 \alpha_j p_j \, c\right) \,\!

in the Heisenberg picture implies that any operator Q obeys the equation

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

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

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

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

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

\alpha_k (t) = \alpha_k (0) e^{-2 i H t / \hbar} %2B c p_k H^{-1}

Then:

x_k(t) = x_k(0) %2B c^2 p_k H^{-1} t %2B {1 \over 2 } i \hbar c H^{-1} ( \alpha_k (0) - c p_k H^{-1} ) ( e^{-2 i H t / \hbar } - 1 ) \,\!

where x_k(t) \,\! 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 of positive- (or entirely of negative-) 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.

See also

References and notes

  1. ^ "Quantum physics: Trapped ion set to quiver". Nature News and Views. http://www.nature.com/nature/journal/v463/n7277/full/463037a.html. 
  2. ^ Gerritsma; Kirchmair, Zähringer, Solano, Blatt, Roos (2010). Nature. http://www.nature.com/nature/journal/v463/n7277/full/nature08688.html. 

Further reading

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