Perturbative relativistic modifications to wave-packet dynamics and uncertainty relations in the quantum harmonic oscillator

TL;DR

This paper derives perturbative relativistic corrections to wave-packet dynamics and uncertainty relations in the quantum harmonic oscillator, showing effects become significant at high energies and velocities, potentially observable experimentally.

Contribution

It provides the first analytical expressions for relativistic corrections to wave-packet parameters in the quantum harmonic oscillator at order 1/c^2.

Findings

Relativistic effects are significant at keV energies and 15% speed of light.

Corrections to uncertainty relations reach up to 1%.

Analytic formulas are derived for Gaussian wave packets.

Abstract

We analyze 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 the wave-packet parameters such as the average position, width and the uncertainty relations. For Gaussian wave packets, these corrections admit closed-form analytic expressions at order $1/c^{2}$. When applied to electron wave packets, the results indicate that relativistic effects become non-negligible in the keV scale harmonic confinement energies -- the uncertainty relationship deviation reaches 0.1% to 1% for an electron wave-packet moving at 15% speed of light and confined within 1-10 keV energy range which might be experimentally verifiable.