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Homework Help: 3D Rectangular Scattering Potential

  1. Nov 1, 2011 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant 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].

    3. 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!
  2. jcsd
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