I'm currently studying particles at a potential step of finite width, and am confused with the nature of the eigenfunctions in the 3 regions.
\begin{align} \psi_I =& Ae^{ikx} + Be^{-ikx} \\ \psi_{II} =& Ce^{\alpha x} + De^{-\alpha x} \\ \psi_{III} =& Fe^{ikx} + Ge^{-ikx} \end{align}
I understand both $\psi_I$ and $\psi_{III}$ as they are travelling waves outside of the potential barrier, and that $G=0$, but why is there an exponential growth term within $ψ_{II}$ whenever the probability of the particle existing should only be decaying as the barrier width increases?