Closed form perturbative relativistic modifications to wave-packet dynamics in the quantum harmonic oscillator

Abstract

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.