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

We consider wave packets of free particles with a general energymomentum dispersion relation E(p). The spreading of the wave packet is determined by the velocity v = ∂pE. The position-velocity uncertainty relation ∆x∆v ≥ 1 2 |〈∂2 pE〉| is saturated by minimal uncertainty wave packets Φ(p) = A exp(−αE(p) + βp). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(p) = p/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)/ma as well as for the relativistic dispersion relation E(p) = √ p2 +m2. The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.