Scattering off two centers

[quZz]

Hello to all and each one of you!

I'm a bit confused about solving Shoroedinger equation
$$\nabla^2 \psi + (p^2 - 2mU(\textbf{r})) \psi = 0,$$
for scattering problem
$$\psi(|\textbf{r}|\to \infty) \sim e^{i\textbf{pr}} + f(\theta,\phi) e^{ipr}/r$$
if potential is of the form
$$U(\textbf{r})=V_1(|\textbf{r}|) + V_2(|\textbf{r}-\textbf{a}|).$$

Assuming potentials are not singular and decrease rapidly enough at long distance what is the best choice of coordinates and what are the corresponding initial/boundary conditions for analytical analysis and numerical calculation?

Thanks for any help

 

[jvicens]

!

Hello there! Solving the Schrodinger equation for a scattering problem can be tricky, but with the right approach, it can be done. The first step is to choose an appropriate coordinate system. In this case, it would be best to use spherical coordinates, as the potential is given in terms of the distance from the origin. This will make the equations simpler and more manageable.

Next, we need to consider the initial and boundary conditions. At infinity, the wave function should behave as a plane wave, as given in the forum post. This means that the wave function should have a form of e^{i\textbf{pr}} at large distances. As for the boundary conditions, we need to consider the behavior of the potential at the origin. If the potential is not singular, then the wave function should be continuous at the origin. This means that the wave function and its derivative should be continuous at the origin.

For numerical calculations, we can use a discretization method such as the finite difference method or the finite element method. These methods involve dividing the space into smaller regions and solving the equations in each region. The initial and boundary conditions can be applied to each region separately.

For analytical analysis, we can use the method of separation of variables. This involves assuming a solution of the form \psi(r,\theta,\phi) = R(r)Y(\theta,\phi) and plugging it into the Schrodinger equation. This will lead to a set of differential equations for R(r) and Y(\theta,\phi), which can be solved separately.

I hope this helps! Let me know if you have any further questions. Good luck with your research!