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

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.

Figure 1

Calculated velocity of wave packet in the absence of external fields for three packet widths $d$. The phenomenon of transient Zitterbewegung is seen. The initial packet wave vector is ${\mathit{k}}_0=(0,0,k_{0z})$ with $k_{0z}=0.8{\lambda }_c^{-1}$. Time is expressed in $t_c=\hslash /m{c}^2$ units. The initial packet velocity is $v_{0z}=\hslash k_{0z}/m=0.8c$; its final velocity depends on packet parameters.

Figure 2

Time evolution of the wave packet (absolute value) according to the one-dimensional version of Eq. (36). The initial wave packet (thick line) splits into two subpackets moving with different velocities. The thin lines show shapes of subpackets in successive time intervals, $2t_c=2\hslash /\left(m{c}^2\right)$.

Figure 3

Time-dependent velocity components for an ellipsoidal wave packet at various magnetic fields. For (a) low fields, cyclotron motion is obtained; for (b), (c) higher fields, packet velocity includes both cyclotron and Zitterbewegung frequencies. In all cases, the motion decays in time.

Figure 4

Average velocity of the spherical wave packet in a longer time scale. The collapse-revival patterns are seen in the ZB oscillations. The motion has a transient character.

Figure 5

Average packet velocity $\langle {\widehat{v}}_z\left(t\right)\rangle$ in the direction parallel to the magnetic field vs time for four values of $B$. The transition to the nonrelativistic limit is visible. Parameters are the same as those used for Fig. 3.

Figure 6

Classical simulation of the KGE according to Morse and Feshbach [13]. A flexible string is anchored at two points and tension $T$ is applied to each end. The string is also attached to a thin rubber sheet. At instant $t$, the shape of the string is given by $y(x,t)$. There are two forces acting on each element $dx$ of the string: restoring force $F_T$ due to applied tension and elastic force $F_K$ of stretched rubber.

Figure 7

Calculated classical variance of the position of the wave packet propagating according to the KGE: nonoscillating part of variance (dashed line) and total variance (solid line). For string oscillations analyzed in the text, there is ${\lambda }_c^s=4.47$ mm and $t_c^s=2.37\times {10}^{-5}$ s.