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

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 $Δx Δv \geq {1/2} |< \p_p^2 E >|$ 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 $E(p)=-\cos (pa)/m{a}^2$ 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.

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• Non locality and causal evolution in QFT

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