One-Dimensional Scattering Problem

[fpaolini]

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.

 

[eljose79]

Dear fellow scientist,

Thank you for sharing your work on the scattering problem of two particles in one dimension. It is always exciting to see new approaches and solutions to such problems.

Based on the information you have provided, it seems that your approach and calculations are correct. However, there are a few things that could potentially cause the non-zero imaginary part in your result.

Firstly, it is important to note that the potential you have chosen is not a Hermitian operator, which means that it does not satisfy the condition of self-adjointness. This can lead to non-physical results and may be the cause of the non-zero imaginary part in your result. It would be helpful to check if your method and calculations are valid for non-Hermitian operators.

Secondly, it is possible that there may be a mistake in your calculations. I would suggest double-checking your equations and calculations to ensure their accuracy. It is also helpful to compare your results with other known solutions or methods to validate your findings.

Lastly, it may be worth considering the effects of boundary conditions on your calculations. The potential you have chosen has a discontinuity at x = a, which could potentially affect the behavior of your wave function and lead to unexpected results.

In conclusion, I believe that your approach and concepts are correct, but there may be some errors in your calculations or considerations that need to be addressed. I hope this helps and wish you success in your research.