In Shankar exercise 5.4.2 we consider the reflection and transmission of a wavefunction through a delta potential $V(x) = aV_0\delta(x)$. I understand that to mean that the potential is zero everywhere except for the origin, where it is infinite.
Intuitively, I thought that an incoming quanton should not transmit through the potential, but I was ready to write myself off as trying to force a classical interpretation on the problem. Quantum tunneling exists, after all. So there will be a transmitted wave.
However, I have trouble reconciling this with my understanding of the infinite square well. In the infinite square well, the quanton's wavefunctions are resonant modes of the potential barriers (which are infinite). Thus, the solutions to the infinite square well - which have periodic boundary conditions - must require that the wave not be able to transmit through the infinite potential. Otherwise, the quanton is not trapped and we cannot have resonant modes.
These two situations indicate conflicting pieces of information. The former implies that particles can transmit through a delta potential, while the latter case implies the opposite. Is there a contradiction here, or have I made a mistake in my understanding of the concepts of QM?