Zitterbewegung
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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 
, or approximately 1.6×1021 Hz.
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[edit] Derivation
The time-dependent Schrödinger equation
where 
is the Dirac Hamiltonian for an electron in free space
implies that any operator Q obeys the equation
![-i \hbar \frac{\partial Q}{\partial t} (t)= \left[ H , Q \right] \,\!\;.](../../../../math/3/f/b/3fbaf435b78be451cf7a0044da23970b.png)
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 \,\!\;.](../../../../math/2/9/d/29d00e97ed6b713bc36837a114d491d2.png)
The above equation shows that the operator αk 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 \,\!\;.](../../../../math/7/3/7/7371fe4d2c288910601235e48b7f7ef3.png)
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:
where 
is the position operator at time 
.
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] See 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.
[edit] References and notes
[edit] Further reading
- 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)
[edit] External links
- The Zitterbewegung Interpretation of Quantum Mechanics, an alternative explanation in addition to positive-negative energy states interference.
- Zitterbewegung in New Scientist
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