Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.47 Born differential cross section
This note derives the Born differential cross section of addendum
{A.30}.
The general idea is to approximate (A.228) for large distances
. Then the asymptotic constant in
(A.216) can be identified, which gives the differential
cross section according to (A.218). Note that the Born
approximation took the asymptotic constant equal
to one for simplicity.
The main difficulty in approximating (A.228) for large
distances is the argument of the exponential in the fraction. It
is not accurate enough to just say that is
approximately equal to . You need the more accurate
approximation
The final approximation is from taking a factor out of the
square root and then approximating the rest by a Taylor series. Note
that the fraction in the final term is the unit vector in
the direction.
It follows that
Also, in the second exponential, since
,
Writing out the complete expression (A.228) and comparing
with (A.216) gives the constant and hence
the differential cross section.