  We derive analytic expressions of the recursive solutions to the
Schr\"{o}dinger's equation by means of a cutoff potential technique for
one-dimensional piecewise constant potentials. These solutions provide a method
for accurately determining the transmission probabilities as well as the wave
function in both classically accessible region and inaccessible region for any
barrier potentials. It is also shown that the energy eigenvalues and the wave
functions of bound states can be obtained for potential-well structures by
exploiting this method. Calculational results of illustrative examples are
shown in order to verify this method for treating barrier and potential-well
problems.
