= Angular scattering probability from complex amplitudes
{title2=$d\sigma_i/d\Omega=(v_i/v_a)|f_{ia}|^2$}
Take <nonrelativistic quantum mechanics>, a short-range interaction, a fixed heavy target, and a common projectile <mass> $m$. An incoming <plane wave> in atomic state $a$ has unit coefficient. In an open <scattering channel> $i$, the target finishes in a normalized atomic state $\chi_i$, and the projectile has positive <velocity> magnitude $v_i=\hbar k_i/m$. At large distance,
$$
\Psi\sim e^{ik_a z}\chi_a+\sum_{i\,\mathrm{open}}f_{ia}(\Omega)\frac{e^{ik_i r}}r\chi_i,\qquad
\frac{\hbar^2k_i^2}{2m}+\varepsilon_i=\frac{\hbar^2k_a^2}{2m}+\varepsilon_a.
$$
Here $\Omega$ denotes direction, $d\Omega$ is a <solid angle> element, and $\varepsilon_i$ is an internal atomic <energy>. Different $\chi_i$ are <orthogonal>. The <scattering amplitude> $f_{ia}$ has units of length. A closed <scattering channel> has no propagating outgoing <probability current> at this total <energy>.
The incoming <probability current> density is $j_{\mathrm{in}}=v_a$. The outgoing radial <probability current> density in channel $i$ is $j_{r,i}\sim v_i|f_{ia}|^2/r^2$. Multiplying by the area $r^2d\Omega$ and dividing by $j_{\mathrm{in}}$ gives the <differential scattering cross-section>:
$$
\frac{d\sigma_i}{d\Omega}=\frac{v_i}{v_a}|f_{ia}(\Omega)|^2.
$$
The $r^{-2}$ dilution cancels the detector's area growth. The <velocity> ratio remains: a squared wave coefficient alone is not generally a flux fraction. A <differential scattering cross-section> has units of area per <solid angle>, rather than being a normalized directional <probability density>. For selected scattering events in acceptance $D$, an ideal detector with equal efficiency in all retained channels instead gives
$$
\Pr(i,\Omega\in D\mid\text{selected scattering})
=\frac{\displaystyle\int_D(v_i/v_a)|f_{ia}|^2\,d\Omega}
{\displaystyle\sum_j\int_{\mathrm{selection}}(v_j/v_a)|f_{ja}|^2\,d\Omega}.
$$
The denominator specifies the event selection; it need not equal one square unit of area. Absolute rates require incident flux, target number and detector efficiency in addition to these relative weights. The next finite-mode model normalizes outgoing <probability current> to one incoming event and predicts event fractions directly.
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