- #1
THEODORE D SAUYET
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I'm preparing for an exam and I expect this or a similar question to be on it, but I'm running into problems with using the Born approximation and optical theorem for scattering off of a finite well.
1. Homework Statement
Calculate the cross sectional area σ for low energy scattering off of a finite well of depth V0 and width a.
Definition of f using the T-matrix:
[tex]f(k,k') = (\frac{2m}{\hbar^2})(\frac{-1}{4\pi})(2\pi)^3<k'|T|k>[/tex]
Where the <k'| and |k> refer to incoming plane waves, I believe.
Optical theorem:
[tex]\sigma_{tot} = \frac{4\pi}{k}\textrm{Im}(f(k,k))[/tex]
And in the Born approximation T ##\approx## V
[/B]
To me, the matrix element ##<k|V|k>## should just be V0a, because
[tex]\int_0^ae^{-ikx}V_0e^{ikx} = V_0a[/tex]
Where the bounds are from 0 to a because the potential is zero outside of the range 0 to a. But then when you plug this into the optical theorem there is no imaginary component, so you get that the total cross section is zero.
Am I missing something about how to properly use the Born approximation/optical theorem? Is this what we expect? Am I doing something wrong in the math? I'm pretty skeptical of how I treated the <k'| and |k>, but it seems fairly consistent, because <k|##V_0##|k> would just be ##V_0##, since the k's are orthonormal.
Any help/insights would be much appreciated!
1. Homework Statement
Calculate the cross sectional area σ for low energy scattering off of a finite well of depth V0 and width a.
Homework Equations
Definition of f using the T-matrix:
[tex]f(k,k') = (\frac{2m}{\hbar^2})(\frac{-1}{4\pi})(2\pi)^3<k'|T|k>[/tex]
Where the <k'| and |k> refer to incoming plane waves, I believe.
Optical theorem:
[tex]\sigma_{tot} = \frac{4\pi}{k}\textrm{Im}(f(k,k))[/tex]
And in the Born approximation T ##\approx## V
The Attempt at a Solution
[/B]
To me, the matrix element ##<k|V|k>## should just be V0a, because
[tex]\int_0^ae^{-ikx}V_0e^{ikx} = V_0a[/tex]
Where the bounds are from 0 to a because the potential is zero outside of the range 0 to a. But then when you plug this into the optical theorem there is no imaginary component, so you get that the total cross section is zero.
Am I missing something about how to properly use the Born approximation/optical theorem? Is this what we expect? Am I doing something wrong in the math? I'm pretty skeptical of how I treated the <k'| and |k>, but it seems fairly consistent, because <k|##V_0##|k> would just be ##V_0##, since the k's are orthonormal.
Any help/insights would be much appreciated!