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I One-Dimensional Scattering Problem

  1. Dec 26, 2016 #1
    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
    \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 C0 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.
     
  2. jcsd
  3. Dec 31, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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