We derive closed form expressions of weak relativistic corrections to the wave-packet dynamics of the quantum harmonic oscillator within a perturbative framework. General expressions are derived for the leading-order relativistic contributions to wave-packet parameters, such as the time-dependent widths, variances, and uncertainty relations. Specific calculations are performed for unsqueezed, minimum-uncertainty Gaussian wave packets, valid up to leading order in ( $1/c^{2}$ ). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach $0.15\% - 1.5\%$ for an electron wave packet confined within the $1-10 \mathrm{keV}$ energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.We derive closed form expressions of weak relativistic corrections to the wave-packet dynamics of the quantum harmonic oscillator within a perturbative framework. General expressions are derived for the leading-order relativistic contributions to wave-packet parameters, such as the time-dependent widths, variances, and uncertainty relations. Specific calculations are performed for unsqueezed, minimum-uncertainty Gaussian wave packets, valid up to leading order in ($1/c^{2}$). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach $0.15\% - 1.5\%$ for an electron wave packet confined within the $1-10 \mathrm{keV}$ energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.

quant-pharXiv:2603.07954Journal of Physics A Mathematical and TheoreticalMar 10, 20260 citations

Electron Particles as Waves

ScienceMar 4, 19320 citations

Vortex-Enhanced Zitterbewegung in Relativistic Electron Wave Packets

Zhongze Guo, Bei Xu, Qiang Gu

Zitterbewegung (ZBW), the trembling motion predicted by the Dirac equation, has long remained unobservable in free electrons due to its sub-Compton scale. We elaborately construct a relativistic vortex electron wave packet as a coherent superposition of both positive- and negative-energy Dirac states and derive their space-time dynamics. Our analysis demonstrates that introducing orbital angular momentum provides a mechanism for amplifying the ZBW amplitude far beyond that of conventional Gaussian packets, while maintaining coherence. The resulting relativistic vortex states unify Gaussian and Bessel-Gaussian models within a single framework and opens new possibilities for observing relativistic quantum dynamics in structured electron wave packets.Zitterbewegung (ZBW), the trembling motion predicted by the Dirac equation, has long remained unobservable in free electrons due to its sub-Compton scale. We elaborately construct a relativistic vortex electron wave packet as a coherent superposition of both positive- and negative-energy Dirac states and derive their space-time dynamics. Our analysis demonstrates that introducing orbital angular momentum provides a mechanism for amplifying the ZBW amplitude far beyond that of conventional Gaussian packets, while maintaining coherence. The resulting relativistic vortex states unify Gaussian and Bessel-Gaussian models within a single framework and opens new possibilities for observing relativistic quantum dynamics in structured electron wave packets.

quant-pharXiv:2511.21142Nov 27, 20250 citations

Electron Particles as Waves

ScienceMar 4, 19320 citations

1/c^{2}

). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach

0.15\% - 1.5\%

for an electron wave packet confined within the

1-10 \mathrm{keV}

energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.