  The exponential decay law is well established since its first derivation in
1928, however it is not exact but only an approximate description. In recent
years some experimental and theoretical indications for non-exponential decay
have been documented. First we solve analytically the time-dependent
Schr\"odinger equation in one dimension for a potential consisting of an
infinite wall plus a rectangular barrier with finite width and also a cut
harmonic oscillator potential by considering it as a sequence of square
potentials. Then using the staggered Leap-Frog method, we solve the
time-dependent Schr\"odinger equation for the cut harmonic oscillator
potential. In both methods, time dependence of the survival probability of the
particle and the decay parameter {\lambda} are analyzed. The results exhibit
non-exponential behavior for survival probability at short and intermediate
times.
