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

摘要

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 = partial derivative E-P. The position-velocity uncertainty relation Delta x Delta V >= 1/2 vertical bar artial derivative E-2(P)>vertical bar 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)/ma(2) as well as for the relativistic dispersion relation E(p) = root 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.