There are three matlab files here. leftside.m and center.m sum linear combinations of eigenstates to produce wave functions that are nonzero on the leftside (0 < x < 0.5) and the center (0.5 - Delta < x < 0.5 + Delta), respectively. As more terms are kept, the series converges to the exact answer. The other m file takes the same wavefunction as in leftside.m and time evolves it. You can see the probability, |psi|^2, oscillate between the left and right sides of the infinite square well.