Zitterbewegung

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Zitterbewegung (English: "jitter") is a theoretical helical or circular motion of elementary particles, in particular electrons, which is responsible for producing their spin and magnetic moment. 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.

[edit] Derivation

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

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

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

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

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

The time-dependence of the velocity operator is given by

$\hbar \frac{\partial \alpha_k}{\partial t} (t)= i\left[ H , \alpha_k \right] = 2ip_k-2i\alpha_kH \,\!\;.$

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:

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

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. Thus, we arrive at the interpretation of the "Zitterbewegung" as being caused by interference between positive- and negative-energy wave components.

[edit] References

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)