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 $k$. We prove that at large times, the wave function is approximately equal to the superposition of two wave packets traveling in opposite directions, which results in trajectories with approximately constant asymptotic momentum $k$ and asymptotic energy $\pm c^2\sqrt{m^2+k^2}$, with $m$ the rest mass of the particle and $c$ the speed of light. 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.