Minimal Position-Velocity Uncertainty Wave Packets in Relativistic and Non-relativistic Quantum Mechanics

TL;DR

This paper investigates minimal position-velocity uncertainty wave packets across various dispersion relations, including non-relativistic, lattice, and relativistic cases, analyzing their spreading, boost properties, and behavior in expanding universes.

Contribution

It derives explicit forms of minimal uncertainty wave packets for different dispersion relations and explores their physical properties and implications.

Findings

Minimal uncertainty wave packets saturate the position-velocity uncertainty relation.

Explicit wave packet forms are obtained for non-relativistic, lattice, and relativistic cases.

Analysis includes boost transformations and wave packet propagation in expanding universes.

Abstract

We consider wave packets of free particles with a general energy-momentum dispersion relation $E(p)$. The spreading of the wave packet is determined by the velocity $v = \p_p E$. The position-velocity uncertainty relation $\Delta x \Delta v \geq {1/2} |< \p_p^2 E >|$ is saturated by minimal uncertainty wave packets $\Phi(p) = A \exp(- \alpha E(p) + \beta p)$. In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation $E(p) = p^2/2m$, analytic calculations are presented for the spreading of wave packets with minimal position-velocity uncertainty product for the lattice dispersion relation $E(p) = - \cos(p a)/m a^2$ as well as for the relativistic dispersion relation $E(p) = \sqrt{p^2 + m^2}$. The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.