  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 $o(s^{-\nu})$ with $\frac{1}{2}<\nu<1$ as the
randomness strength $s$ increases. This suggests an onset of localization, but
it is weaker than Anderson localization.
