Much of the lore of the Zitterbewegung of a free Dirac particle is found to need revision for the situation when the particle is initially localized. We find that in this case the rapid violent oscillations of angular frequency ∠2mc2/h/ in the average position of the particle are a transient rather than sustained phenomenon. Instead, the Zitterbewegung is manifested by a growth in the size of the particle but this growth is many orders of magnitude smaller than h//mc, the magnitude usually quoted. This behavior is contrasted with that of the Foldy-Wouthuysen position for which the Zitterbewegung is absent. Finally, the applicability of these ideas to the situation of a localized Dirac particle in an external field is commented on.
Zitterbewegung (ZB) oscillation, namely, the jittering center-of-mass motion predicted by free-space Dirac (or Dirac-like) equations, has been studied in several different contexts. It is shown here that ZB can be converted to directed center-of-mass motion by a modulation of the Dirac-like equation, if the modulation is on resonance with the ZB frequency. Tailored modulation may also stop, re-launch or even reverse the directed motion of a wavepacket with negligible distortion. The predictions may be examined by current ZB experiments using trapped-ion systems.
After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time
trev, at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than
trev. We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period
tsr≫trev with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time
trev. Furthermore, at times that are rational fractions of
tsr, we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time
trev. These periodicities indicate a new type of fractional revival, occurring for times much greater than
trev. We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.
In this work we study the effects of collapse and revival, as well as the zitterbewegung (ZB) phenomenon, for the relativistic electron wave packets, which are a superposition of the states with quantum numbers sharply peaked around some Landau level
n0 of the order of few tens. The probability densities as well as average velocities of the packet center and the average spin components were calculated analytically and their evolution is visualized. Our computations demonstrate that due to the dephasing of the states for times larger than the cyclotron period the initial wave packet (which includes the states with the positive energy only) loses the spatial localization so that the evolution can no longer be described classically. However, at the half-revival time
t=TR/2 its reshaping takes place first. It is shown that the behavior of the wave packet containing the states of both energy bands (with
En>0 and
En<0) is more complicated. At short times of a few classical periods such a packet splits into two parts, which rotate with cyclotron frequency in the opposite directions and meet each other every one-half of the cyclotron period. At these moments their wave functions have a significant overlap that leads to ZB, and at the time of fractional revival each of two subpackets is decomposed into few packet fractions. However, at
t=TR each of the two subpackets (with positive or negative energy) restores at various points of the cyclotron orbit, which makes reshaping of the initial wave packet impossible, entirely unlike the wave packet which consists of states with energies
En>0 only. The obtained results can be useful for the description of electromagnetic radiation and absorption in relativistic plasma on astrophysics objects, where superhigh magnetic field has a value of the order
108−−109T, as well as for interpretation of experiments with trapped ions.
We seek the {\em immediate} description of chiral oscillations in terms of the trembling motion described by the velocity (Dirac) operator {\boldmath
α}. 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.
In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.