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arXiv:1205.4707 · Physical Review A

Zitterbewegungof Klein-Gordon particles and its simulation by classical systems

Tomasz M. Rusin, Wlodek Zawadzki

Published Sep 7, 2012• 4 citations
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Abstract

The Klein-Gordon equation is used to calculate the Zitterbewegung (ZB, trembling motion) of spin-zero particles in the absence of fields and in the presence of an external magnetic field. Both Hamiltonian and wave formalisms are employed to describe ZB and their results are compared. It is demonstrated that if one uses wave packets to represent particles, then the ZB motion has a decaying behavior. It is also shown that the trembling motion is caused by an interference of two subpackets composed of positive- and negative-energy states, which propagate with different velocities. In the presence of a magnetic field, the quantization of the energy spectrum results in many interband frequencies contributing to ZB oscillations and the motion follows a collapse-revival pattern. In the limit of nonrelativistic velocities, the interband ZB components vanish and the motion is reduced to cyclotron oscillations. The exact dynamics of a charged Klein-Gordon (KG) particle in the presence of a magnetic field is described on an operator level. The trembling motion of a KG particle in the absence of fields is simulated using a classical model proposed by Morse and Feshbach---it is shown that a variance of a Gaussian wave packet exhibits ZB oscillations.

DOI
10.1103/physreva.86.032103

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The Klein-Gordon equation is used to calculate the Zitterbewegung (ZB, trembling motion) of spin-zero particles in the absence of fields and in the presence of an external magnetic field. Both Hamiltonian and wave formalisms are employed to describe ZB and their results are compared. It is demonstrated that if one uses wave packets to represent particles, then the ZB motion has a decaying behavior. It is also shown that the trembling motion is caused by an interference of two subpackets composed of positive- and negative-energy states, which propagate with different velocities. In the presence of a magnetic field, the quantization of the energy spectrum results in many interband frequencies contributing to ZB oscillations and the motion follows a collapse-revival pattern. In the limit of nonrelativistic velocities, the interband ZB components vanish and the motion is reduced to cyclotron oscillations. The exact dynamics of a charged Klein-Gordon (KG) particle in the presence of a magnetic field is described on an operator level. The trembling motion of a KG particle in the absence of fields is simulated using a classical model proposed by Morse and Feshbach---it is shown that a variance of a Gaussian wave packet exhibits ZB oscillations.
Zitterbewegung (ZB, the trembling motion) of free relativistic electrons in a vacuum in the presence of an external magnetic field is calculated. It is shown that the motion of an electron wave packet has intraband frequency components, corresponding to the classical cyclotron motion, and several interband frequency components corresponding to the Zitterbewegung. For a two-dimensional situation, the presence of a magnetic field makes the ZB motion stationary, i.e. not decaying in time. We show how to simulate the ZB in a magnetic field using trapped ions and laser excitations in the spirit of recently observed proof-of-principle ZB simulation by Gerritsma {\it et al.} Nature {\bf 463}, 68 (2010). It is demonstrated that, for the parameters of the Dirac equation simulated by the above experiment, the effect of a magnetic field on the Zitterbewegung is considerable.
One-electron 3+13+13+1 and 2+12+12+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion, Zitterbewegung (ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron ZB: it introduces intraband frequency components into the motion, influences all the frequencies, and makes the motion stationary (not decaying in time) in case of the 2+12+12+1 Dirac equation. Finally, simulations of the 2+12+12+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers, we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.
The Hamiltonian form of the Klein–Gordon equation is used to study the Zitterbewegung of spin- zero particles. It is shown that the time dependence of the velocity and position operators in the Heisenberg picture is formally the same as that of a spin- 1/2 particle. It is also shown that the Zitterbewegung is the result of the interference between positive and negative energy states. A representation-free expression is found for the mean position operator. This operator is free from Zitterbewegung. The Klein–Gordon equation is solved with an electrostatic potential originally used by Sauter in his investigation of the Klein paradox for spin- 1/2 particles. It is shown that the transmission to states with negative kinetic energy is very small when the electric field is weak.
We seek the {\em immediate} description of chiral oscillations in terms of the trembling motion described by the velocity (Dirac) operator {\boldmathααα}. By taking into account the complete set of Dirac equation solutions which results in a free propagating Dirac wave packet composed by positive and negative frequency components, we report about the well-established {\em zitterbewegung} results and indicate how chiral oscillations can be expressed in terms of the well know quantum oscillating variables. We conclude with the interpretation of chiral oscillations as position very rapid oscillation projections onto the longitudinally decomposed direction of the motion.
Zitterbewegung of massless particles with an arbitrary spin is analyzed in various representations. Dynamics of the group velocity of a massless particle as a whole and of the corresponding radius vector is determined. This radius vector defines any fixed point of the envelope of the moving wave packet characterizing the particle and its group velocity differs from the group velocities of any points of the wavefront. We consider free massless scalar and Dirac particles, the photon, and massive and massless particles with an arbitrary spin and describe them in different representations. For particles with an arbitrary spin, the generalized Feshbach-Villars representation and the Foldy-Wouthuysen one are used. Zitterbewegung takes place in any representation except for the Foldy-Wouthuysen one. Formulas describing the motion of a ``trembling'' free particle are the same in any representation. In the Foldy-Wouthuysen representation, the operators of the velocity and momentum of a free particle are proportional and Zitterbewegung does not take place. Since the radius vector (position) and velocity operators are the quantum-mechanical counterparts of the classical position and velocity just in the Foldy-Wouthuysen representation, Zitterbewegung is not observable. The same conclusion has been previously made for free massive particles. For relativistic massive particles with spins 0, 1/2, 1 and massless particles with spins 0, 1/2 in arbitrarily strong electromagnetic fields, independent of the external fields Zitterbewegung does not appear in the Foldy-Wouthuysen representation either. This conclusion is made for leading terms in the Hamiltonian proportional to the zero and first powers of the Planck constant and for such terms proportional to ${\ensuremath{\hbar}}^{2}$ which describe contact interactions.
We explore the dynamics of relativistic quantum waves in a potential step by using an exact solution to the Klein-Gordon equation with a point-source initial condition. We show that in both the propagation and Klein-tunneling regimes, the zitterbewegung effect manifests itself as a series of quantum beats of the particle density in the long-time limit. We demonstrate that the beating phenomenon is characterized by the zitterbewegung frequency and that the amplitude of these oscillations decays as ${t}^{\ensuremath{-}3/2}$. We show that the beating effect also manifests itself in the free Klein-Gordon and Dirac equations within a quantum shutter setup, which involve the dynamics of cutoff quantum states. We also find a time domain where the particle density of the point source is governed by the propagation of a main wavefront, exhibiting an oscillating pattern similar to the diffraction-in-time phenomenon observed in nonrelativistic systems. The relative positions of these wavefronts are used to investigate the time delay of quantum waves in the Klein-tunneling regime. We show that depending on the energy difference E\mathcal{E}E between the source and the potential step, the time delay can be positive, negative, or zero. The latter case corresponds to a super-Klein-tunneling configuration, where E\mathcal{E}E equals half the energy of the potential step.