Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics

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 E . The position-velocity uncertainty relation Δ x Δ v ⩾ 1 2 | 〈 ∂ p 2 E 〉 | is saturated by minimal uncertainty wave packets Φ ( p ) = A exp ( - α E ( p ) + β p ) . In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E ( p ) = p 2 / 2 m , 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 2 as well as for the relativistic dispersion relation E ( p ) = 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.