It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting “two-horned” distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of “coin” and “walker” are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac’s equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the $x$ axis for the Dirac particle at any time $t$ can be obtained from the asymptotic form of the probability distribution for the position of a “quantum walker.” The case of a highly localized initial state is discussed as an example.

• Received 22 May 2006

DOI:https://doi.org/10.1103/PhysRevA.75.022322