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.

$$ψ_I = Ae^{(ikx)} + Be^{(-ikx)}$$

$$ψ_{II} = Ce^{(αx)} + De^{(-αx)}$$

$$ψ_{III} = Fe^{(ikx)} + Ge^{(-ikx)}$$

I understand both $ψ_I$ and $ψ_{III}$ 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?

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?

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.

$$ψI = Aexp(ikx) + Bexp(-ikx)$$

$$ψII = Cexp(αx) + Dexp(-αx)$$

$$ψIII = Fexp(ikx) + Gexp(-ikx)$$

I understand both $ψI$ and $ψIII$ as they are travelling waves outside of the potential barrier, and that $G=0$.

But why is there an exponential growth term within $ψII$ whenever the probability of the particle existing should only be decaying as the barrier width increases?

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.

$$ψ_I = Ae^{(ikx)} + Be^{(-ikx)}$$

$$ψ_{II} = Ce^{(αx)} + De^{(-αx)}$$

$$ψ_{III} = Fe^{(ikx)} + Ge^{(-ikx)}$$

I understand both $ψ_I$ and $ψ_{III}$ 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?

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.

$ψI = Aexp(ikx) + Bexp(-ikx)$

$ψII = Cexp(αx) + Dexp(-αx)$

$ψIII = Fexp(ikx) + Gexp(-ikx)$

I understand both $ψI$ and $ψIII$ as they are travelling waves outside of the potential barrier, and that G=0.

But why is there an exponential growth term within $ψII$ whenever the probability of the particle existing should only be decaying as the barrier width increases?

Thank you for your help!

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.

$$ψI = Aexp(ikx) + Bexp(-ikx)$$

$$ψII = Cexp(αx) + Dexp(-αx)$$

$$ψIII = Fexp(ikx) + Gexp(-ikx)$$

I understand both $ψI$ and $ψIII$ as they are travelling waves outside of the potential barrier, and that $G=0$.

But why is there an exponential growth term within $ψII$ whenever the probability of the particle existing should only be decaying as the barrier width increases?