  We use both continuum and lattice models to study the energy-momentum
dispersion and the dynamics of a wave packet for an electron moving in graphene
in the presence of spin-orbit couplings and either a single potential barrier
or a periodic array of potential barriers. Both Kane-Mele and Rashba spin-orbit
couplings are considered. A number of special things occur when the Kane-Mele
and Rashba couplings are equal in magnitude. In the absence of a potential, the
dispersion then consists of both massless Dirac and massive Dirac states. A
periodic potential is known to generate additional Dirac points; we show that
spin-orbit couplings generally open gaps at all those points, but if the two
spin-orbit couplings are equal, some of the Dirac points remain gapless. We
show that the massless and massive states respond differently to a potential
barrier; the massless states transmit perfectly through the barrier at normal
incidence while the massive states reflect from it. In the presence of a single
potential barrier, we show that there are states localized along the barrier.
Finally, we study the time evolution of a wave packet in the presence of a
periodic potential. We discover special points in momentum space where there is
almost no spreading of a wave packet; there are six such points in graphene
when the spin-orbit couplings are absent.
