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

الأصل · EN

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 Δx Δv ≥ 1/2 |< ₚ² E >| is saturated by minimal uncertainty wave packets Φ(p) = A (- αE(p) + βp). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(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) = - (p a)/m a² 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.