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One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion ofa relativistic electron in a vacuum in the presence of an external magneticfield. First, calculations are carried on an operator level and exactanalytical results are obtained for the electron trajectories which containboth intraband frequency components, identified as the cyclotron motion, aswell as interband frequency components, identified as the trembling motion(Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used forthe same problem to compute average values of electron position and velocityemploying Gaussian wave packets. It is shown that the presence of a magneticfield and the resulting quantization of the energy spectrum has pronouncedeffects on the electron Zitterbewegung: it introduces intraband frequencycomponents into the motion, influences all the frequencies and makes the motionstationary (not decaying in time) in case of the 2+1 Dirac equation. Finally,simulations of the 2+1 Dirac equation and the resulting electron ZB in thepresence of a magnetic field are proposed and described employing trapped ionsand laser excitations. Using simulation parameters achieved in recentexperiments of Gerritsma and coworkers we show that the effects of thesimulated magnetic field on ZB are considerable and can certainly be observed.

Zitterbewegung of relativistic electrons in a magnetic field and its simulation by trapped ions