  We show how the Laplace transform can be used to give a solution of the
time-dependent Schr\"odinger equation for an arbitrary initial wave packet if
the solution of the stationary equation is known. The solution is obtained
without summing up eigenstates nor do we need the path integral. We solve the
initial value problem for three characteristic piecewise constant potentials.
The results give an intuitive picture of the wave packet dynamics, reproducing
explicitly all possible reflection and transmission processes. We investigate
classical and quantum properties of the evolution and determine the reflection
and transmission probabilities.
