As shown recently [H. F. Hofmann, Phys. Rev. A 96, 020101(R) (2017)], it is possible to demonstrate that quantum particles do not move along straight lines in free space by increasing the probability of finding the particles within narrow intervals of position and momentum beyond the “either/or” limit of 0.5 using constructive quantum interference between a component localized in position and a component localized in momentum. The probability of finding the particle in a corresponding spatial interval at a later time then violates the lower bound of the particle propagation inequality which is based on the validity of Newton's first law. In this paper, the problem of localizing the two state components in their respective target intervals is addressed by introducing a set of three coefficients that describe the localization of arbitrary wave functions quantitatively. This characterization is applied to a superposition of Gaussians, obtaining a violation of the particle propagation inequality by more than 5% if the width of the Gaussian wave function is optimized along with the size of the position and momentum intervals. It is shown that the violation of the particle propagation inequality originates from the fundamental way in which quantum interferences relate initial position and momentum to the future positions of a particle, indicating that the violation is a fundamental feature of causality in the quantum limit.

• Received 28 June 2018
• Revised 27 September 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052104