Solve the time-independent Schrödinger equation for a centered infinite square well with a delta-function barrier in the middle:
V(x)={αδ(x),∞,−a<x<+a,∣x∣≥a.
Treat the even and odd wave functions separately. Don't bother to normalize them. Find the allowed energies (graphically, if necessary). How do they compare with the corresponding energies in the absence of the delta function? Explain why the odd solutions are not affected by the delta function. Comment on the limiting cases α→0 and α→∞.