Past exam of the theoretical physics II course of the University of Cambridge / 2017 / 4 / a / Solution

Outside a short-range scattering region, the stationary Schrödinger equation reduces to the two-dimensional Helmholtz equation. Separation in polar coordinates gives outgoing radial Hankel functions, whose large-$r$ dependence is $r^{-1/2}{e}^{ikr}$. The same power follows from conserved probability current: a circle has length proportional to $r$, so the squared outgoing amplitude must decrease as $1/r$.

For an incident plane wave in the $x$ direction, adopt the dimensionless amplitude conventionThe phase $e^{i\pi /4}$ and factor $k^{-1/2}$ specify the definition of $f$; they can be absorbed into a differently normalized scattering amplitude. Here they are kept exactly so that the later two-dimensional scattering cross-section follows consistently.