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)E(p). The spreading of the wave packet is determined by the velocity v=∂pEv=∂pE. The position-velocity uncertainty relation ΔxΔv⩾12|〈∂p2E〉| is saturated by minimal uncertainty wave packets Φ(p)=Aexp(-αE(p)+βp)Φ(p)=Aexp(-αE(p)+βp). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(p)=p2/2mE(p)=p2/2m, 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)/ma2E(p)=-cos(pa)/ma2 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.