The negative flow of probability Available to Purchase

We derive an inequality that the parameters of a 1D free-particle Gaussian wave packet with a positive group velocity, approaching a given region x > q, must satisfy such that a negative probability current J exists on q. Local probability conservation implies the counter-intuitive result that the particle detection probability in the region x > q is actually decreasing. The condition J < 0 requires the negative correlation of the position and momentum observables of the state, but the time scales for the negative current and anti-correlation regimes are not identical. Using a probability current operator, we obtain an integral representation of J in momentum space for any free particle wave packet. We use this integral representation to distinguish the separate contributions to J by the positive and negative momentum components, and we identify a third contribution to J composed of cross-terms of both momenta. For the specific case of a Gaussian wave packet with a negative correlation between its position and momentum, the positive momentum component can contribute a negative value to the probability current.