  In this paper, we study discrete-time quantum walks on one-dimensional
lattices. We find that the coherent dynamics depends on the initial states and
coin parameters. For infinite size of lattice, we derive an explicit expression
for the return probability, which shows scaling behavior $P(0,t)\sim t^{-1}$
and does not depends on the initial states of the walk. In the long-time limit,
the probability distribution shows various patterns, depending on the initial
states, coin parameters and the lattice size. The average mixing time
$M_{\epsilon}$ closes to the limiting probability in linear $N$ (size of the
lattice) for large values of thresholds $\epsilon$. Finally, we introduce
another kind of quantum walk on infinite or even-numbered size of lattices, and
show that the walk is equivalent to the traditional quantum walk with
symmetrical initial state and coin parameter.
