3D Rectangular Scattering Potential

[qmphysics]

Homework Statement

There is a constant potential, V0, in the region -a<x<a, -a<y<a, -b<z<b, and V=0 otherwise. Particles of mass m are incident on the scatterer with wave vector k in the x direction with a flux at the origin of one particle per second per cm^2. There is a detector with cross-sectional area A is located at a large distance R from the origin, in the direction with polar coordinates (theta, phi). Assumptions: a<<b, k*a<<1, V0<<hbar^2/(m*b^2), k*b not necessarily small. We want to find the counting rate in the detector. The problem also asks how the answer would change if the k vector was parallel to z instead.

Homework Equations

It seems that due to the small potential, we can use the Born approximation, which gives the scattering amplitude [itex]f(k\prime,k)=-\frac{m}{2 \pi \hbar^2}\int exp(i (k-k\prime)\cdot r\prime) V(r\prime) d^{3}r\prime)[/itex].

The Attempt at a Solution

In spherical coordinates, the Born approximation isn't bad to calculate. However, in this rectangular case, I can't get a nice looking result out of the integration above. I'm interested to see if anyone thinks there is a different way to do the problem, or if I should just keep trying the integral. As for the second part of the problem, it doesn't seem like the Born approximation would necessarily hold when the particle sees a longer range potential. But hopefully this part will become more clear when I can figure out the first part.

Thanks for your help!

 

[mark726]

Thank you for your post. You are correct in using the Born approximation for this problem, as the potential is small and the incident wave vector is small compared to the size of the scatterer. However, in this case, using rectangular coordinates may not yield a simple solution for the integral. You may want to try using spherical coordinates instead, as the potential is spherically symmetric. This may give you a simpler expression for the scattering amplitude, which can then be used to calculate the counting rate at the detector.

As for the second part of the problem, you are correct in thinking that the Born approximation may not hold when the incident wave vector is parallel to the z-axis. In this case, the potential will have a larger effect on the scattering and a different approach may be needed. You may want to consider using the full Schrödinger equation to solve for the scattering amplitude in this case.

I hope this helps and good luck with your calculations! If you need any further assistance, please don't hesitate to ask. As scientists, we are always here to help and support each other in our research and problem-solving.