# One-Dimensional Scattering Problem

• I
• fpaolini
Your Name]In summary, the conversation discusses a scattering problem of two particles in one dimension, utilizing the T operator theory and a chosen potential. The chosen solution and calculations result in a non-zero imaginary part, which could be caused by the non-Hermitian nature of the potential, calculation errors, or boundary conditions. The ideas and concepts are likely correct, but further examination and validation of the calculations may be necessary.
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

U(x) = \left\{ {\begin{array}{cc}
U_0 & 0 < x < a \\ 0 & \rm{Otherwise} \ \end{array} }\right.

I have choice the solution in such a way that

\lim_{x\rightarrow\,\infty} \psi \rightarrow\, C_0\exp{i\kappa\,x}
Where x is the relative position, $$\hbar\,\kappa$$ is the relative momentum and C0 is a normalization constant. Once given the wave function solution I have calculated the matrix element

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

The matrix element above if put in integral form is

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

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.

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.

## 1. What is the one-dimensional scattering problem?

The one-dimensional scattering problem is a mathematical model used to describe the behavior of waves or particles as they interact with a potential barrier or potential well in one dimension. It is commonly used in physics and engineering to study the transmission and reflection of waves or particles in various systems.

## 2. What are the key assumptions made in the one-dimensional scattering problem?

The one-dimensional scattering problem assumes that the potential barrier or well is constant in space and time, and that the incident wave or particle is a plane wave with a well-defined wavelength. It also assumes that there is no interaction between the waves or particles, and that the potential barrier or well is infinite in height and width.

## 3. What is the role of boundary conditions in the one-dimensional scattering problem?

Boundary conditions play a critical role in the one-dimensional scattering problem as they determine the behavior of the wave or particle at the potential barrier or well. They specify how the wave or particle behaves at the boundaries of the potential, and are essential in solving the mathematical equations that describe the scattering process.

## 4. How is the one-dimensional scattering problem solved?

The one-dimensional scattering problem can be solved using various mathematical methods such as the Schrödinger equation, the wave equation, or the transfer matrix method. These methods involve solving differential equations and applying boundary conditions to determine the transmission and reflection coefficients of the scattering process.

## 5. What are some applications of the one-dimensional scattering problem?

The one-dimensional scattering problem has a wide range of applications in different fields such as quantum mechanics, acoustics, and electromagnetics. It is used to study the behavior of particles in particle accelerators, the transmission of sound waves through barriers, and the interaction of light with different materials. It is also used in the design of devices such as waveguides, optical fibers, and quantum dots.

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