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.Comment: 12 pages, 8 figure