- 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 byErwin Schrödinger in1930 as a result of his analysis of thewave packet solutions of theDirac equation for relativistic electrons in free space, in which aninterference between positive and negativeenergy state s 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 (mathbf{x},t) = i hbar frac{partialpsi}{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 ight) ,!
implies that any operator Q obeys the equation
:i hbar frac{partial Q}{partial t} (t)= left [ H , Q ight] ,!;.
In particular, the time-dependence of the position operator is given by
:hbar frac{partial x_k}{partial t} (t)= ileft [ H , x_k ight] = 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)= ileft [ H , alpha_k ight] = 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 tofind the explicit time-dependence of the position operator. First:
:alpha_k (t) = alpha_k (0) e^{-2 i H t / hbar} + c p_k H^{-1}
Then:
: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 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 thezero-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
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)
External links
* [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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