The Dirac equation is the fundamental equation of relativistic quantum mechanics. While it improves on certain aspects of the Schrödinger equation it has solutions that are rather difficult to interpret.
It is a crucial advantage of the free Dirac equation that it is invariant with respect to Lorentz transformations and hence it fulfills one of the basic requirements of the special theory of relativity.
This movie shows a velocity transformation of a Gaussian wave packet in one dimension. It does not show a time evolution, but just the result of observing the wave packet out of a moving inertial frame. The relative velocity between wave packet and observer is plotted above the image. The effect of Lorentz contraction is clearly visible: The wave packet appears to be narrower if viewed from a fast moving frame of reference. But the most interesting phenomenon is the development of an interference pattern due to the presence of negative energies in the wave packet. We refer to "Visual Quantum Mechanics-Book Two" for a more detailed explanation.