I have tried to solve a scattering problem of two particles in one dimension, following the T operator theory, after to write the system in the center of mass reference. I have used the square potential(adsbygoogle = window.adsbygoogle || []).push({});

\begin{equation}

U(x) = \left\{ {\begin{array}{cc}

U_0 & 0 < x < a \\ 0 & \rm{Otherwise} \ \end{array} }\right.

\end{equation}

I have choice the solution in such a way that

\begin{equation}

\lim_{x\rightarrow\,\infty} \psi \rightarrow\, C_0\exp{i\kappa\,x}

\end{equation} Where x is the relative position, \begin{equation} \hbar\,\kappa\end{equation} is the relative momentum and C_{0}is a normalization constant. Once given the wave function solution I have calculated the matrix element

\begin{equation}

t(k\leftarrow\,k) = \left<k|U|\psi_k\right> = \left<k\right|\hat{T}\left|k\right>

\end{equation}

The matrix element above if put in integral form is

\begin{equation}

\left<k\right|\hat{T}\left|k\right> = \int_0^a\,\exp{\left(-i\kappa\,x\right)}\,U_0\,\psi_k(x)dx

\end{equation}

I have done this calculation but the result should give a pure real number, however my result provides a non zero imaginary part.

Someone could point me out if at leat the ideas above are right and if not which is wrong?

I do not know if there is some error in the concepts or it is just error during calculations.

Thanks.

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# I One-Dimensional Scattering Problem

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