# How to tell when exponentials are real valued? (Barrier Potential)

Where $k_1=\frac{\sqrt{2mE}}{\hbar}$ and $\alpha=\frac{\sqrt{2m(V_0-E)}}{\hbar}$

I'm quite confused as to why the exponentials in regions I and III are complex functions while in region II the exponentials are real valued. I'm pretty sure in region II it's because $\alpha>0$ although this doesn't appear to be a good explanation since $k_1$ is also positive.

In region (1) since $V=0$ the SE becomes

$$\frac{h^2}{2m}\frac{\partial^2 \psi}{\partial x^2}=-E\psi$$

since $E>0$ we always need $k$ to be real so we take $k=\frac{\sqrt{2mE}}{\hbar}$

This will give solutions of the form $\psi=Ae^{ikx}+Be^{-ikx}$

In region(2) SE becomes $$\frac{h^2}{2m}\frac{\partial^2 \psi}{\partial x^2}=-(E-V_0)\psi$$

but observe that $E<V_0$ so we cannot define $k=\frac{\sqrt{2m(E-V_0)}}{\hbar}$ since k will become an imaginary number, so we have to define

$k=\frac{\sqrt{2m(V_0-E)}}{\hbar}$

Which will give solutions of the kind $\psi=Ae^{kx}+Be^{-kx}$