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?

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

If you set your potential to be a complex number, your energy operator is no more hermitian and could have complex eigenvalues. The Schrödinger equation tells you that energy is the propagator of time. If the energy is no more hermitian, so the time propagator is no more unitary and you don't have just a phase due to time evolution in an energy eigenstate. This means that probability is no more conserved.

$$\langle\psi|U^\dagger(t) U(t)|\psi\rangle\neq\langle\psi|\psi\rangle$$ 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?


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