What Is the Transmission Coefficient for a Rectangular Barrier?

[Damascus Road]

Question:

Determine the transmission coefficient for a rectangular barrier. Treat separately the three cases E<0, E=Vo, E>Vo (note that the wave function inside the barrier is different in the three cases.
and
V(x) = Vo > 0 for -a < x < a
= 0 for |x| > a



I attempted to do this like the finite square well and do it per "zone", because I'm not sure how to go about dealing with the energy first.

If I call - infinity -> -a "zone 1" and the potential there is zero, the general solution should be:
$$\varphi$$ = A$$e^{-kx}$$ + B$$e^{kx}$$, but the first term blows up to -infinity, so we're left with
$$\varphi$$ = B$$e^{kx}$$.

Can someone tell me what is wrong with this logic?
For E<Vo and x < -a $$\varphi$$ should include both A and B and be imaginary, but I don't know why.

 

[Damascus Road]

I'm sure it's apparent that this is from a textbook, (Griffiths 2.33) but it is study, not homework. Please don't disregard on that basis.

 

[Damascus Road]

To generalize a little bit more, this is is my basic question... why is the incoming wave to a square barrier complex?

 

[Damascus Road]

Nevermind.