  We survey the equations of continuous-time quantum walks on simple
one-dimensional lattices, which include the finite and infinite lines and the
finite cycle, and compare them with the classical continuous-time Markov
chains. The focus of our expository article is on analyzing these processes
using the Laplace transform on the stochastic recurrences. The resulting time
evolution equations, classical versus quantum, are strikingly similar in form,
although dissimilar in behavior. We also provide comparisons with analyses
performed using spectral methods.
