= 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> $f_k(\theta,\phi)$:
$$
\boxed{\psi_k(\mathbf r)\sim e^{ikz}+f_k(\theta,\phi)\frac{e^{ikr}}r.}
$$
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 $(\hbar k/m)|f_k|^2/r^2$ to leading order, while the incident current density is $\hbar k/m$. Therefore $d\sigma/d\Omega=|f_k|^2$. The sign $e^{+ikr}$ assumes the stationary <time> factor $e^{-iE_kt/\hbar}$.
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