Abstract

The phenomenon of probability backflow, previously quantified for a free nonrelativistic particle, is considered for a free particle obeying Dirac's equation. It is shown that probability backflow can occur in the opposite direction to the momentum; that is to say, there exist positive-energy states in which the particle certainly has a positive momentum in a given direction, but for which the component of the probability flux vector in that direction is negative. It is shown that the maximum possible amount of probability that can flow “backwards,” over a given time interval of duration T, depends on the dimensionless parameter ε = (4ℏ/mc2T)1/2, where m is the mass of the particle and c is the speed of light. At ε = 0, the nonrelativistic value of approximately 0.039 for this maximum is recovered. Numerical studies suggest that the maximum decreases monotonically as ε increases from 0, and show that it depends on the size of m, ℏ, and T, unlike the nonrelativistic case