 Solution

Outside a finite-range scattering potential, the stationary Schrödinger equation becomes the free Helmholtz equation. The specified incident state is a plane wave along $+z$. The scattered disturbance must be outgoing: the scattering experiment has an incoming incident beam but no independently imposed spherical wave arriving from infinity. At large radius its angular pattern is encoded by a scattering amplitude $f_k(\theta ,\varphi )$:An outgoing three-dimensional wave has amplitude proportional to $1/r$ because its probability current spreads over area proportional to $r^2$. Indeed its radial current is $(\hslash k/m)|f_k{|}^2/r^2$ to leading order, while the incident current density is $\hslash k/m$. Therefore $d\sigma /d\Omega =|f_k{|}^2$. The sign $e^{+ikr}$ assumes the stationary time factor $e^{-i{E}_{k}t/\hslash }$.