Zitterbewegung of Klein-Gordon particles and its simulation by classical systems

TL;DR

This paper investigates the Zitterbewegung (trembling motion) of Klein-Gordon particles using both Hamiltonian and wave formalisms, demonstrating decay, interference effects, and classical simulation of the phenomenon.

Contribution

It provides a comparative analysis of ZB using different formalisms and introduces a classical model to simulate ZB oscillations of Klein-Gordon particles.

Findings

ZB exhibits decay when using wave packets.

Interference of positive and negative energy states causes ZB.

Classical models can simulate ZB oscillations.

Abstract

The Klein-Gordon equation is used to calculate the Zitterbewegung (ZB, trembling motion) of spin-zero particles in 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, the ZB motion has a decaying behavior. It is also shown that the trembling motion is caused by an interference of two sub-packets composed of positive and negative energy states which propagate with different velocities. In the presence of a magnetic field the quantization of energy spectrum results in many interband frequencies contributing to ZB oscillations and the motion follows a collapse-revival pattern. In the limit of non-relativistic velocities the interband ZB components vanish and the motion is reduced to cyclotron oscillations. The exact dynamics of a charged Klein-Gordon particle in the presence of a magnetic field is described on an operator level. The trembling motion of a KG particle in 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.