Asymptotic Momentum of Dirac Particles in One Space Dimension

We analyze the trajectories of a massive particle in one space dimension whose motion is guided by a spin-half wave function that evolves according to the free Dirac equation, with its initial wave function being a Gaussian wave packet with a nonzero expected value of momentum $k$ and the positive expected value of energy $E = \sqrt{m^2+k^2}$. We prove that at large times, the wave function becomes {\em locally} a plane wave, which corresponds to trajectories with fixed values for asymptotic momentum $k$ and asymptotic energy $E$ or $-E$. The sign of the asymptotic energy is determined by the initial position of the particle. Particles with negative energy will have an asymptotic velocity that is in the opposite direction of their momentum. The proof uses the stationary phase approximation method, for which we establish a rigorous error bound.