The mean square displacement of a ballistic quantum particle

Abstract

A commonly used quantum mechanical formulation of the mean square displacement δ x 2 ( t ) is based on the quantum correlation function of the position operator. While this quantity yields the classically expected result δ x 2 ( t ) ∝ t 2 for a ballistic particle in the interval topology, it is found here to diverge at essentially all times, when evaluated on infinitely large rings. A somewhat different formulation of the mean square displacement in quantum mechanics was proposed in a previous work (R. Marquardt, Mol. Phys. 119, e1971315 (2021)) and yielded the result δ x 2 ( t ) ∝ t for the ballistic particle on an infinitely large ring. Here, it is shown analytically that that formulation yields δ x 2 ( t ) ≡ 0 for the ballistic particle on an infinitely long interval. The two formulations define two different, topology dependent quantities that have the dimension of an area and that could in principle be determined from the same experiment, following a measurement idea proposed in the aforementioned work. That idea is critically reviewed here.