Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 11.14.
In classical scattering theory, the simplest approximation is the impulse approximation, in which a particle’s path is assumed to be a straight line right through the scattering region, and the total impulse resulting from the component of force perpendicular to the particle’s trajectory is calculated. This impulse is assumed to be a small fraction of the particle’s incoming horizontal momentum , so the scattering angle should be small, and given approximately by
As an example, we’ll apply the impulse approximation to Rutherford scattering of a charge travelling with kinetic energy from another charge at rest. We assume has an impact parameter (that is, if the particle didn’t interact with the target, it would pass by with a closest approach distance of ).
The impulse is the change in momentum that a force produces over a given time, so the required impulse from the perpendicular component of a force is
We’ll take ‘s straight line trajectory to be along the axis and place at on the axis. Then the Coulomb force between and is
with a perpendicular component of
The impulse is
To convert this to an integral over we can use the fact that ‘s speed is constant at
So we have
where the integral is over the range of from its closest approach when out to infinity. We’ve doubled the integral to account for the incoming () and outgoing () legs of the journey.
The integral evaluates to
The incoming momentum is
The exact answer is
For small , so
which is consistent with 17.