= Two-channel scattering with flux normalization
{title2=$S^\dagger S=I,\quad P_i=(v_i/v_1)|c_i|^2$}
Consider one chosen model at a fixed total <energy>, with two open <scattering channels>, no absorption and no omitted outgoing modes. Channel 1 leaves the target unchanged; channel 2 excites it. Let $v_1=v>0$ and $v_2=v/2$. For the same projectile <mass>, <conservation of energy> then requires a target excitation $\Delta=3mv^2/8$. These are hypothetical parameters, not a fit to a particular atom. Each incoming or outgoing mode is normalized to unit <probability current>; the outgoing modes will have the angular profiles specified below.
Choose the <S-matrix>
$$
S=\frac15\begin{pmatrix}3&4i\\4i&3\end{pmatrix},\qquad
S^\dagger S=\frac1{25}\begin{pmatrix}25&12i-12i\\-12i+12i&25\end{pmatrix}=I.
$$
This phenomenological <unitary matrix> specifies the channel conversion; it is not derived here from an interaction <potential energy>. For incidence in channel 1, its first column gives the outgoing flux amplitudes
$$
b=S\begin{pmatrix}1\\0\end{pmatrix}
=\begin{pmatrix}3/5\\4i/5\end{pmatrix},\qquad
P_1=\frac9{25},\quad P_2=\frac{16}{25}.
$$
To recover ordinary wave coefficients relative to an incident coefficient one, undo the <velocity> normalization:
$$
c_i=\sqrt{\frac{v_1}{v_i}}b_i,\qquad
c_1=\frac35,\quad c_2=\frac{4i\sqrt2}{5},\qquad
\sum_i\frac{v_i}{v_1}|c_i|^2=\frac9{25}+\frac12\frac{32}{25}=1.
$$
Thus $|c_2|^2=32/25>1$ is allowed: channel 2 travels more slowly. Its outgoing <probability current> fraction is $16/25$, not $32/25$. Renormalizing the unweighted squares would give $9/41$ and $32/41$, the wrong <probabilities>. More generally the ordinary coefficient map is $A=V^{-1/2}SV^{1/2}$, with $V=\operatorname{diag}(v,v/2)$; it obeys $A^\dagger VA=V$. The conserved quantity is the squared coefficient <norm> weighted by <velocity>, which represents <probability current>.
Now specify an original angular realization of these two outgoing modes, with <scattering angle> $\theta$ and no azimuth dependence:
$$
g_1(\theta)=\sqrt{\frac3{16\pi}}(1+i\cos\theta),\qquad
g_2(\theta)=\frac1{\sqrt{4\pi}},\qquad
\int|g_1|^2d\Omega=\int|g_2|^2d\Omega=1.
$$
The first normalization follows from $2\pi\int_{-1}^1(1+u^2)\,du=16\pi/3$, where $u=\cos\theta$. These angular profiles define the modes of this model; they are not asserted to follow from Born's approximation or an atomic <potential energy>. The joint directional <probability densities> are
$$
\frac{dP_1}{d\Omega}=\frac9{25}\frac3{16\pi}(1+\cos^2\theta),\qquad
\frac{dP_2}{d\Omega}=\frac{16}{25}\frac1{4\pi}.
$$
\Image[/born-scattering-complex-amplitude-angular-intensity.png]
{title=Complex angular amplitudes become channel-resolved scattering intensity}
{description=Original two-channel model. Left: arrows in the <complex plane> show $h(\theta)=1+i\cos\theta$, the angular factor of $g_1$. Right: the joint directional <probability densities> $P_i|g_i|^2$, per <solid angle>, retain the channel flux weights $9/25$ and $16/25$. The shaded interval is the cone $0\leq\theta\leq60^\circ$; its <probability> requires the measure $2\pi\sin\theta\,d\theta$, not the area under a curve against degrees. These are chosen amplitudes, not historical measurements.}
{source=born-scattering-complex-amplitude-angular-intensity.py}
Suppose a detector accepts the cone $D:0\leq\theta\leq\pi/3$, with full azimuth and unit efficiency. Its angular fractions within the channels are
$$
F_1=\frac38\int_{1/2}^1(1+u^2)\,du=\frac{19}{64},\qquad
F_2=\frac12\int_{1/2}^1du=\frac14.
$$
Multiplying the angular fractions by the outgoing flux weights gives
$$
\Pr(1,D)=\frac9{25}\frac{19}{64}=\frac{171}{1600},\qquad
\Pr(2,D)=\frac{16}{25}\frac14=\frac{256}{1600}.
$$
For $1600$ independent incident events, the <expected values> are therefore $171$ and $256$ accepted detections, compared with $576$ and $1024$ over all directions. The remaining expected $1173$ events fall outside the cone. The chosen incident number makes the expectations integers; actual counts fluctuate. Conditional on a detection inside the cone, the channel <probabilities> are $171/427$ and $256/427$, rather than $9/25$ and $16/25$. Angular acceptance changes the observed channel proportions even with equal detector efficiencies.
\Image[/born-scattering-channel-counts-weighted-flux.png]
{title=Velocity weighting and angular acceptance determine expected channel counts}
{description=Original model with $1600$ incident events. Left: raw squared wave coefficients $9/25$ and $32/25$ become conserved outgoing flux fractions $9/25$ and $16/25$ after multiplying by $v_i/v_1$. Right: all-direction expectations $576$ and $1024$ become $171$ and $256$ inside the $60^\circ$ cone. Bars show calculated <expected values>, not a generated sample or experimental counts.}
{source=born-scattering-channel-counts-weighted-flux.py}
The final atomic states are <orthogonal>, so ignoring the atomic readout adds the two channel intensities; it does not produce $|b_1g_1+b_2g_2|^2$. Coherent paths that reach the same final state can instead interfere, as the existing <probability amplitude> and <quantum superposition> examples demonstrate. The distinction is physical distinguishability of final states, not whether the detector operator chooses to display their labels. Flux normalization and angular integration make the scattering interpretation useful: they turn complex outgoing waves into conserved event weights and then into the statistics of a specified detector.
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