Quantum tunnelling eigenfunctions

Question

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?

Answer 1

It all boils down to boundary conditions here.

This is what the wave function will typically look like--so your intuition about the positive exponential isn't wrong. The reason that term must be included is because you cannot immediately say that the solution to the Schrödinger equation in region (II) doesn't have a positive exponential component (just as you can't say immediately that there are no "backward" traveling waves in region (III)). The positive exponent only exists in a finite region of space so the infinity problem doesn't apply here. After you apply boundary conditions the form of the wave function in region (II) will be more enlightening--the nature of the wave function's probability density here is to decay, never to grow.