Say the particle is traveling to the right and hits the barrier. Before hitting, you have your typical solutions, say Aexp(ikx) + Bexp(-ikx). After hitting, the V(psi) and E(psi) terms cancel. Integrating then yields some Cx + D, which diverges at infinity, so C=0 and you have constant wave function, psi(x)=D

Here's where my question is. You can either ignore normalization and find A,B in terms of D, the compute things like transmission/reflection coefficients. When I did this I got A=B=D/2.

I computed R=(|B|/|A|)^2 = 1, and also T = (|D|/|A|)^2 = 4.

So I normalize these by hand and say R=1/5, T=4/5

On the other hand, a constant wave function is non-normalizable. So what does this mean? My friend thinks it is a non physical solution, and thus all the wave must be reflected. But the free particle is also initially non-normalizable until we express it as an integral over a continuous wave number. But that's not exactly the same thing here, and the non-normalizable argument seems strong.

Our problem was to find the SE for E=V, without normalizing, then find the reflection coefficient.

So basically, what happens when E=V for a barrier potential