Bohmian approach to spin-dependent time of arrival for particles in a uniform field and for particles passing through a barrier

It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin-1/2 particles. In the nonrelativistic domain this implies a guidance law for electrons which differs by an additional spin-dependent term from the one originally proposed by de Broglie and Bohm. Although the additional term in the guidance equation may not be detectable in the quantum measurements derived solely from the probability density $rho$, it plays a role in the case of arrival-time measurements. In this paper we compute the arrival time distribution and the mean arrival time at a given location, with and without the spin contribution, for two problems: 1) a symmetrical Gaussian packet in a uniform field and 2) a symmetrical Gaussian packet passing through a 1D barrier. Using the Runge-Kutta method for integration of the guidance law, Bohmian paths of these problems are also computed.