# Transmission + Reflection coefficients >1 For Potencial Barrier with Negative Complex Part Contradicts Paper

I am studying reflection and transmission coefficients for a barrier consisting of a a step potencial defined by:

$$V(x):=\begin{cases}0&{\rm if}\,|x|>a/2 \\ V_0+iW_0 & {\rm if}\,|x|<a/2\end{cases}$$

The transmission and reflection coefficients are as defined in this paper; using Mathematica I plotted these coefficients in this notebook; my problem is that their sum is above one, for instance, for the following parameters ( I am using as energy unit $\frac{\hbar^2}{2m}$):

$$V_0=1 \\ W_0=-1 \\ a=5 \\ E=0.75$$

From what I have understood from the paper, this shouldn't happen. From what I read from other sources, there are no derivations of $R+T$ being less than one for $W_0<0$, which is what the paper claims. This may be due to me misunderstanding the theory, or me misunderstanding Mathematica. Perhaps I should post it in their stackexchange too?

• Dropbox link now dead. – Qmechanic Mar 1 '16 at 22:13

$$\langle\psi|U^\dagger(t) U(t)|\psi\rangle\neq\langle\psi|\psi\rangle$$ because $$U(t)=\exp\left(\frac{i}{\hbar}Ht\right)$$ and $$H^\dagger \neq H$$
So, if you choose $$H\,|E\rangle= e \,|E\rangle$$ with $$e=a+ib$$ then $$\exp\left(\frac{i}{\hbar}et\right)=\exp\left(\frac{-bt}{\hbar}\right)\left[\cos\left(\frac{at}{\hbar}\right)+i\sin\left(\frac{at}{\hbar}\right)\right]$$ this clearly does not preserve probabilities.
Furthermore,this paper shows that: $$\frac{\langle E|Im(V(\hat{x}))|E\rangle}{\sqrt{E}}= 1-T-R,$$ for $E$ real
I think that your mistake is purely mathematical, perhaps you got the sign of $W_0$ wrong somewhere?