On the Ehrenfest theorem and centroids of relativistic particles

TL;DR

This paper investigates the relativistic Ehrenfest theorem, revealing that mean velocity and momentum are not always aligned in relativistic particles, but the energy centroid's velocity remains aligned with momentum, impacting angular momentum decomposition.

Contribution

It demonstrates that the simple proportionality between velocity and momentum breaks down in relativistic dispersion, and clarifies the behavior of energy centroids in relativistic quantum particles.

Findings

Velocity of energy centroid aligns with mean momentum in relativistic particles.

Proportionality between velocity and momentum holds only for quadratic dispersion.

Implications for angular momentum decomposition in relativistic systems.

Abstract

We consider relativistic versions of the Ehrenfest relation between the expectation values of the coordinate and momentum of a quantum particle in free space: $d\langle {\bf r} \rangle /dt = \langle {\bf p} \rangle/m$. We find that the simple proportionality between the mean velocity and momentum holds true only for the simplest quadratic dispersion (i.e., dependence of the energy on the momentum). For relativistic dispersion, the mean velocity is generally not collinear with the mean momentum, but velocity of the {\it energy centroid} is directed along the mean momentum. This is related to the conservation of the Lorentz-boost momentum and has implications in possible decomposition of the mean orbital angular momentum into intrinsic and extrinsic parts. Neglecting spin/polarization effects, these properties depend solely on the dispersion relation, and can be applied to any waves, including classical electromagnetic or acoustic fields.