= 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 convention
$$
\boxed{\psi(r,\theta)\sim e^{ikr\cos\theta}+\frac{e^{i\pi/4}}{\sqrt{kr}}f(\theta)e^{ikr}.}
$$
The 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.
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