(9041) Scattering by narrow scatterer above a potential floor.
A particle of mass \( M \) moves with energy \( E \) above a potential floor \( E_F \), and is scattered by a potential \( V(r) = u \delta(r) \).
The density of the momentum states \( k \) in energy is \( \nu \text{Volume} \).
The matrix elements of the widened delta potential are
\( V_{k,k'} = u \) for \( |E_k {-} E_{k'}| < \Delta_0 \)
and zero otherwise.
The Born expansion can be written as
\( f(\Omega)=\sum_{n=1}^{\infty} f^{(n)}(\Omega) \).
(1) Write what is \( f^{(1)}(\Omega) \).
What are the [Length] [Time] units of \(M\) and \(u\) and of the result?
(2) Find what is \( f^{(2)}(\Omega) \) if \( E > E_F+\Delta_0 \)
(3) Calculate \( f^{(2)}(\Omega) \) if \( E_F < E < E_F+\Delta_0 \)
(4) Extend the answer to item 2 and find
what is \( f^{(n)}(\Omega) \) if the potential floor can be ignored.
(5) What is the exact result for \( f(\Omega) \) in the latter case?
Verify that the answer makes sense as far as units are concerned.
Tip: It might be more convenient to use box normalized \( k \) states,
such that the normalization is \( \langle k|k'\rangle = \delta_{k,k'} \).
With such normalization \( V_{k,k'} = u / \text{Volume} \),
Born fomula for \( f(\Omega) \) aquires an extra \( \text{Volume} \) factor,
and the summation measure is \( \sum_k = \text{Volume} \int\nu dE \).
Please verify that your final result has the correct units...