Arrival/Detection Time of Dirac Particles in One Space Dimension

In this paper we study the arrival/detection time of Dirac particles in one space dimension. We consider particles emanating from a source point inside an interval in space and passing through detectors situated at the endpoints of the interval that register their arrival time. Unambiguous measurements of “arrival time” or “detection time” are problematic in the orthodox narratives of quantum mechanics, since time is not a self-adjoint operator. We instead use an absorbing boundary condition proposed by Tumulka for Dirac’s equation for the particle, which is meant to simulate the interaction of the particle with the detectors. By finding an explicit solution, we prove that the initial-boundary value problem for Dirac’s equation satisfied by the wave function is globally well-posed, the solution is smooth, and depends smoothly on the initial data. We verify that the absorbing boundary condition gives rise to a non-negative probability density function for arrival/detection time computed from the flux of the conserved Dirac current. By contrast, the free evolution of the wave function (i.e., if no boundary condition is assumed) will not in general give rise to a nonnegative density, while Wigner’s proposal for arrival time distribution fails to give a normalized density when no boundary condition is assumed. As a consistency check, we verify numerically that the arrival time statistics of Bohmian trajectories match the probability distribution for particle detection time derived from the absorbing boundary condition.