Localization of relativistic particles have been of great research interests over many decades. We investigate the time evolution of the Gaussian wave packets governed by the one dimensional Dirac equation. For the free Dirac equation, we obtain the evolution profiles analytically in many approximation regimes, and numerical simulations consistent with other numerical schemes. Interesting behaviors such as Zitterbewegung and Klein paradox are exhibited. In particular, the dispersion rate as a function of mass is calculated, and it yields an interesting result that super-massive and massless particles both exhibit no dispersion in free space. For the Dirac equation with random potential or mass, we employ the Chebyshev polynomials expansion of the propagator operator to numerically investigate the probability profiles of the displacement distribution when the potential or mass is uniformly distributed. We observe that the widths of the Gaussian wave packets decrease approximately with the power law of order $o(s^{- u})$ with $frac{1}{2}< u<1$ as the randomness strength $s$ increases. This suggests an onset of localization, but it is weaker than Anderson localization.
Download