Minimal length effect on the broadening of free wave packets and its physical implications
We study the Generalized Uncertainty Principle (GUP) modified time evolution for the width of wave-packets for a scalar potential. Free particle case is solved exactly where the wave-packet broadening is modified by a coupling between the GUP parameter and higher order moments in the probability distribution in momentum space. We consider two popular forms of deformations widely used in the literature - one of which modifies the commutator with a quadratic term in momentum, while the other modifies it with terms both linear and quadratic in momentum. Unlike the standard case, satisfying Heisenberg uncertainty, here the GUP modified broadening rates, for both deformations, not only depend on the initial size (both in position and momentum space) of the wave-packet, but also on the initial probability distribution and momentum of the particle. The new rates of wave-packet broadening, for both situations, are modified by a handful of new terms - such as the skewness and kurtosis coefficients, as well as the (constant) momentum of the particle. Comparisons with the standard Heisenberg Uncertainty Principle (HUP)-based results show potentially measurable differences in the rates of free wave-packet broadening for physical systems such as the $C_{60}$ and $C_{176}$ molecules, and more so for large organic molecular wave-packets. In doing so, we open a path to scan the GUP parameter space by several orders of magnitude inside the best existing upper bounds for both forms of GUP.