A novel mechanism for probing the Planck scale with wave packets following general distributions

The generalized uncertainty principle (GUP) is predicted by most theories of quantum gravity and in turn introduces a minimum measurable length in nature. It was also shown recently that GUP predicts potentially measurable corrections to the "doubling time" of freely moving Gaussian atomic and molecular wave packets with a favorable combination of three parameters, e.g., mass, initial width and mean velocity of a traveling wave packet. However, it is well known that such wave packets can come with various shapes which correspond to a variety of distributions. In this article, we generalize our earlier work for an arbitrary distribution and thereby accommodate any shape of the wave packet. Mathematically, we build this formalism by exploiting a duality between quantum and statistical mechanics, by which (quantum mechanical) expectation values of the momentum operator can be expressed in terms of the derivatives of the characteristic functions of the dual statistical description. Equipped with this result, we go one step further and numerically study a few physical distributions. We find that large organic (TPPF152) wave packet following the generalized normal distribution with parameter κ=0.5 offers one of the best-case scenarios, effectively scanning the whole GUP parameter space with current technologies. Although we do not say that the minimal length has to be near or at the Planck value, we manage improving our previous studies to scan the minimal length signatures down to hundred times the Planck value.