The Zitterbewegung of a Dirac Electron in a Central Field

Alexei A. Deriglazov

James L. Beck

S.V. Vonsovskii

Yanrong Wang, Jie Cao, Sang Xiong

Tomasz M. Rusin, Wlodek Zawadzki

One-electron $3+1$ and $2+1$ 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 $2+1$ Dirac equation. Finally, simulations of the $2+1$ 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.

James H. Wilson

The fact that the Dirac equation predicts Zitterbewegung (ZBW) for a free electron is well known from Schrodinger [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] from the very early days of Dirac's relativistic quantum theory. Schrodingerdescribed ZBW qualitatively as a persistent interference between positive and negative energy states, and the objective of this and previous papers by this author [Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] is to describe ways to measure the presence or absence ofZBW for a free (or nearly free) electron. If ZBW measurements show that it is not present for a free electron, then the Dirac Equation and Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] are inaccurate. If ZBW ispresent as Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] of the Dirac Equation asserts, then a description of a free electron emerges that is significantly different than the present very small, static point electronwith “intrinsic” properties of spin and magnetic moment accepted by most physicists [Eides et al., Phys. Reports 342, 63 (2001).] today. This paper develops a quantitative theory of the free electron directly from the Dirac Equation and Schrodinger's analyses[E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] without any other assumptions. The fact that ZBW exists for the electron in the hydrogen atom is confirmed by the Darwin term [J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley,Reading, MA, 1967)]. It seems unnatural to the author that ZBW would vanish when the electron is removed from the hydrogen atom, and becomes free. The unanswered question is whether ZBW can be confirmed or refuted by measurement for a free electron, and possible ways to measure its presenceor absence are described in this paper. The remainder of the paper describes the free electron, assuming ZBW exists. The Dirac Equation [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] produces a physical description of a free electron's ZBW as rapidly oscillating,quantized Center of Charge (CoC) point rotating about a fictitious Center of Mass (CoM) at a radius of 3/ 2 times the electron's Compton wavelength. The results in the remainder of this paper, as well as Schrodinger's analyses [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.)24, 418 (1930).], are only true if ZBW can be confirmed by measurement. In case measurements show no ZBW exists for a free electron, the Dirac Equation is not a correct description of a single free electron, and more serious problems arise. The static point like electron deduced fromscattering experiments [Eides et al., Phys. Rep. 342, 63 (2001)] that give estimates of the free electron's internal structure based on far field measurements can be misleading in estimating the free electron's internal structure. In Classical Electrodynamics, when one observesthe electric field from a point electron and from a sphere with the same charge spread uniformly over its surface from the far field, Gauss' Law shows that the electric fields are identical. When one tries to “observe” or “measure” the free electron's internal structurewith the high-energy photons required, the single particle free electron is completely destroyed by the measurement process. In fact, any attempt to observe or measure a free electron's structure ensures that the electron is no longer a totally free single particle, and one must be carefulto use only elastic observation methods [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] for this very difficult measurement. Observing a free Quark, much less its internal structure, has proven very difficult, if not impossible.