Asymptotic Form of the Scattering Wave Function

Every article in this chapter leans on one piece of furniture: far from the target, the scattering state looks like $e^{ikz}+f(\theta,\phi)\,e^{ikr}/r$ . Scattering Amplitude defines its central object on top of that form; Lippmann-Schwinger Equation builds it into the boundary condition; every measured cross section is read through it. This article is where the furniture is actually built — and where its small print is measured rather than mumbled. Four results are proved here. First, the asymptotic ansatz theorem: outside the potential's range the Schrödinger equation becomes the free Helmholtz equation, whose radial solutions are $e^{\pm ikr}/r$ — and the $1/r$ is not an approximation but the only falloff that conserves probability flux (amplitude $r^{-p}$ gives flux $\propto r^{2-2p}$ ; verified symbolically). Second, the outgoing-selection theorem: the radial current of $e^{-ikr}/r$ points inward, so demanding that the target emit rather than absorb picks $e^{+ikr}$ — Sommerfeld's radiation condition, tied to the causal taxonomy of Retarded and Advanced Green Functions. Third, the range ladder: how fast $V(r)$ must die, with the honest Coulomb failure — a logarithmically distorted phase — stated with its source. Fourth, the owned laboratory: the far-zone onset theorem. For a source of size $d$ the form switches on at $r\sim kd^2$ , the Fraunhofer distance — derived from the dropped quadratic phase, then measured on a two-emitter toy: error falling exactly as $1/r$ , the $10\%$ knee at $12.9\,d$ for $kd=10$ , and thermal-neutron experiments sitting nine orders of magnitude inside asymptopia.

Key takeaways

Far from a short-range potential every stationary scattering state at $E=\hbar^2k^2/2m$ takes the universal form $\psi\to e^{ikz}+f(\theta,\phi)\,e^{ikr}/r$ .
The $1/r$ is forced, not chosen: an amplitude falling as $r^{-p}$ carries flux $\propto r^{2-2p}$ through spheres, and only $p=1$ conserves probability (machine-verified).
Outgoing means outgoing: $j_r[e^{+ikr}/r]=+(\hbar k/m)|A|^2/r^2$ while $e^{-ikr}/r$ carries the same current inward — the boundary condition that selects the physical state; the $+i\epsilon$ prescription in disguise (Retarded and Advanced Green Functions).
Validity needs $\int^\infty V\,dr<\infty$ . Bare Coulomb fails logarithmically — the phase becomes $kr-\eta\ln 2kr$ (stated with source; screened rescue → Scattering by a Screened Coulomb Potential).
Owned and measured: the far zone begins at the Fraunhofer distance $r\sim kd^2$ — error $\simeq kd^2/(8r)$ , slope $-1$ measured, $10\%$ knee at $r_{10}=12.9\,d$ for $kd=10$ .
The detector direction is the final momentum: $\mathbf k'=k\hat{\mathbf r}$ — pointing a counter is choosing $\mathbf k'$ .
Key idea: far from a short-range scatterer the energy eigenstate becomes $\psi\to e^{ikz}+f(\theta,\phi)\,e^{ikr}/r$ — the outgoing $1/r$ wave is fixed by three-dimensional flux conservation, and everything a detector can measure rides on its angular amplitude.

The core insight

Far from a short-range scatterer the energy eigenstate becomes $\psi\to e^{ikz}+f(\theta,\phi)\,e^{ikr}/r$ ; the outgoing $1/r$ spherical wave is fixed by 3D flux conservation, the outgoing choice is fixed by the direction of its probability current, and the form switches on beyond the Fraunhofer distance $r\sim kd^2$ .

Where this lives today

Every scattering facility is an applied far-field theorem. Neutron and X-ray instruments quote momentum transfer per detector pixel on the strength of $\mathbf k'=k\hat{\mathbf r}$ , with sample-to-detector distances chosen so even mesoscopic samples stay safely Fraunhofer. Antenna engineering runs the same arithmetic in reverse: the "far-field range" $2D^2/\lambda$ stamped on every anechoic-chamber test plan is this article's $kd^2$ in optics conventions. The near zone became its own instrument: electron ptychography and coherent diffractive imaging deliberately record inside the Fresnel regime — the quadratic phase this article drops is their signal, inverted computationally to reconstruct targets at sub-ångström resolution.

The physical picture

Zoom out from any scattering event and three nested regions appear. Innermost, the interaction zone ( $r\lesssim R$ ): $V\neq0$ , the wave function complicated, target-specific, invisible to a distant experimenter. Next, the near zone: the potential is already zero but the outgoing ripple still remembers the target's shape — waves from different parts arrive with different phases, and no single "amplitude times $e^{ikr}/r$ " describes them. Finally, past the Fraunhofer distance $r\sim kd^2$ , the far zone: internal path differences have collapsed below a radian, the target has shrunk to a point with a directional voice, and the universal form takes over. The experimental miracle is how early "infinity" arrives. A thermal neutron ( $\lambda=1.81$ Å) on an atom ( $d\sim1$ Å) has $kd^2\approx3\times10^{-10}$ m — a third of a nanometre, against a detector at $30$ cm: nine powers of ten inside asymptopia. But the scale is physical, not rhetorical: shine $500$ nm light on a millimetre object and $kd^2\approx13$ m — the far field hasn't started by the end of the bench, which is why laser labs need lenses to reach the Fraunhofer regime. Inside the far zone the geometry does one more favour: the wave crossing a detector at direction $\hat{\mathbf r}$ is locally a plane wave of momentum $\hbar k\hat{\mathbf r}$ — the detector's position reads out the final momentum, $\mathbf k'=k\hat{\mathbf r}$ . Counting at an angle is a momentum measurement.

Figure

The three zones: interaction, near field, and the far zone where the universal form lives

Figure

Punchline — asymptopia has a doorstep and we measured it: two-emitter error vs distance, slope −1, knee at r ≈ 1.3 kd²

Free space starts early: the Helmholtz reduction

The construction rests on one observation: the wave function only needs to be simple where the detector is, and the detector sits where $V=0$ . There, the time-independent Schrödinger equation at $E=\hbar^2k^2/2m$ becomes

\left(\nabla^2+k^2\right)\psi=0 \qquad (1)

where: $k=\sqrt{2mE}/\hbar$ is the beam wavenumber — the free Helmholtz equation. Outside the target, every scattering state at the same energy obeys the same equation, whatever the potential did inside; targets differ only through the boundary data they hand outward, which is why one universal form serves every short-range potential in the chapter. (Separating (1) in spherical coordinates is the Spherical Coordinates and the Helmholtz Equation story in this article's prerequisites; The Free Particle in Three Dimensions owns the free spherical solutions.) For the radial factor, substitute $u(r)=\chi(r)/r$ : the radial Helmholtz operator collapses to $\chi''+k^2\chi=0$ (verified symbolically), whose solutions are $\chi=e^{\pm ikr}$ . Dividing back:

u_\pm(r)=\frac{e^{\pm ikr}}{r} \qquad (2)

where: $u_+$ marches outward and $u_-$ is its inward mirror. Both solve (1) exactly for $r>0$ (machine-checked). At $r=0$ they fail usefully: the operator acting on $e^{ikr}/4\pi r$ produces $-\delta^3(\mathbf r)$ — not this article's business but the entire business of Green's Function for Scattering.

The asymptotic ansatz, derived

Derivation AThe asymptotic ansatz theorem: why plane wave plus f times outgoing spherical wave

We are deriving the universal far-field form: that any stationary scattering state approaches $e^{ikz}+f(\theta,\phi)\,e^{ikr}/r$ , with the $1/r$ power forced by probability conservation rather than assumed. The strategy has three moves: split the state into "what was sent in" plus "what the target added"; show the added piece must ride $e^{ikr}$ with some angular modulation; then let the flux integral kill every radial power except $1/r$ . The step to watch is the last one — most treatments assert the $1/r$ ; here it is the unique survivor of a one-line calculation.

Step 1. Split the state. Write $\psi=\psi_{\mathrm{beam}}+\psi_{\mathrm{sc}}$ : the solution with the target removed, plus a remainder. Both $\psi$ and $e^{ikz}$ solve (1) in the exterior, so $\psi_{\mathrm{sc}}$ does too — and unlike the beam it must consist, far out, of waves leaving the target (sharpened in the next derivation). Why the plane wave is consistent furniture. $e^{ikz}$ contains both incoming and outgoing spherical components — the Rayleigh split is Partial Wave Expansion of Plane Waves' theorem, recalled in one line: the beam's own outgoing half is part of "no scattering," so $f$ measures only the excess the target adds.

Step 2. Outgoing waves with an angular voice. The most general exterior outgoing solution has, at large $r$ , the leading form

\psi_{\mathrm{sc}}\;\longrightarrow\;f(\theta,\phi)\,\frac{e^{ikr}}{r^{\,p}}

for some falloff power $p$ : the radial phase must be $e^{ikr}$ (energy fixes $k$ ; outgoing fixes the sign), while the prefactor may depend on direction — the target radiates differently into different angles.

Step 3. Flux conservation selects p = 1. Compute the radial probability current of that trial wave and integrate over a sphere of radius $r$ . With amplitude $\propto r^{-p}$ ,

\oint \mathbf j\cdot d\mathbf A \;=\; \frac{4\pi\hbar k}{m}\,|f|^2_{\mathrm{avg}}\; r^{\,2-2p}

Why this is the executioner. Probability streaming outward through every sphere must be the same — nothing accumulates in empty space in a stationary state. $r^{2-2p}=\mathrm{const}$ forces $p=1$ , exactly: faster falloff would mean probability vanishing en route, slower would mean empty space manufacturing it. (Machine-verified symbolically.)

Step 4. Assemble.

\psi(\mathbf r)\;\xrightarrow{\;r\to\infty\;}\;e^{ikz}\;+\;f(\theta,\phi)\,\frac{e^{ikr}}{r} \qquad (3)

The remaining coefficient $f(\theta,\phi)$ — one complex length per direction — is the scattering amplitude, and its story (dimensions, $\mathrm d\sigma/\mathrm d\Omega=|f|^2$ , the interference audit) is Scattering Amplitude's, built directly on this theorem. Verified: $(\nabla^2+k^2)e^{\pm ikr}/r=0$ for $r>0$ ; flux through spheres $=4\pi(\hbar k/m)|A|^2$ independent of $r$ at $p=1$ and $\propto r^{2-2p}$ otherwise; $u=\chi/r$ reduction exact — 18-check suite green.

Outgoing means outgoing: the selection theorem

Equation (2) offered two exact exterior waves, $e^{+ikr}/r$ and $e^{-ikr}/r$ , and nothing in the time-independent equation prefers one: complex-conjugating a solution gives a solution at the same energy. The choice is physics, not mathematics — and it is worth one short derivation, because it is the same choice that reappears as $+i\epsilon$ in every Green function of the chapter.

Go deeper · Derivation B · 1 minThe outgoing-selection theorem and the radiation condition

Derivation BThe outgoing-selection theorem and the radiation condition

We are proving that the boundary condition "the target emits" translates into keeping $e^{+ikr}/r$ and discarding $e^{-ikr}/r$ — by computing where each wave's probability current actually points. The route is a two-line current calculation; the payoff is the license to call (3) the scattering state rather than a solution. Watch the sign: it is the entire content of the theorem.

Step 1. Current of the outgoing candidate. For $\psi=A\,e^{+ikr}/r$ , the radial component of $\mathbf j=(\hbar/m)\,\mathrm{Im}(\psi^*\nabla\psi)$ evaluates to

j_r\big[e^{+ikr}/r\big]=+\frac{\hbar k}{m}\,\frac{|A|^2}{r^2}

positive everywhere: probability streams away from the origin, $4\pi(\hbar k/m)|A|^2$ of it per second through every sphere. (The $1/r$ -prefactor differentiation contributes only a real term, killed by $\mathrm{Im}$ — the current is exactly this, no corrections.)

Step 2. Current of the incoming candidate. The same computation for $e^{-ikr}/r$ flips one sign:

j_r\big[e^{-ikr}/r\big]=-\frac{\hbar k}{m}\,\frac{|A|^2}{r^2}

Why this settles it. A wave converging on the target forever is legitimate mathematics — the time reverse of scattering — but an experiment prepares a beam and then finds waves leaving; causality lives in the boundary condition. Demanding $j_r>0$ for the scattered piece selects $e^{+ikr}$ : Sommerfeld's radiation condition (1912), stated precisely to make exterior problems unique.

Step 3. The chapter-wide echo. Outgoing-at-infinity is retarded-in-time: Retarded and Advanced Green Functions' table pairs $\hat G^+$ ( $E+i\epsilon$ ) with "outgoing $e^{+ikR}$ … scattering out-states." When Lippmann-Schwinger Equation writes $(E-H_0\pm i\epsilon)^{-1}$ and Green's Function for Scattering evaluates the kernel, the $\pm$ is exactly the sign chosen here by hand. One boundary condition, three dialects. Verified: both radial currents symbolic — $\pm(\hbar k/m)|A|^2/r^2$ exactly; the discarded solution is the complex conjugate (time reverse) of the kept one.

How fast must the potential die? The range ladder

The exterior argument assumed "outside the range, $V=0$ ." Real potentials have tails, and the asymptotic form survives only if the tail's accumulated effect on the phase converges. A wave crossing the tail picks up phase at a rate $\propto V(r)/\hbar v$ ; the accumulated distortion is finite exactly when

\int^{\infty}\! V(r)\,\mathrm{d}r \;<\;\infty \qquad (4)

where: the integral runs over the tail. For $V\sim r^{-n}$ the ladder reads: $n=2$ gives $\int^\infty r^{-2}dr=1/R$ — converges, asymptotic form safe; even $n=3/2$ converges ( $2/\sqrt R$ ); but $n=1$ — the Coulomb tail — gives $\int^\infty dr/r$ , a logarithmic divergence (all three verified symbolically). The wave never stops accumulating phase.

The Coulomb honesty card. For $V=Z_1Z_2e^2/r$ the failure is precise: the asymptotic phase is distorted to $kr-\eta\ln 2kr$ , Sommerfeld parameter $\eta=Z_1Z_2e^2m/\hbar^2k$ — both waves carry the logarithm forever (Landau & Lifshitz Vol. 3, Ch. XVII; stated, not derived). Physical rescue: real charges are screened, $V\sim e^{-\mu r}/r$ kills the tail, and the screened problem — including how Rutherford's formula survives the limit — is owned by Scattering by a Screened Coulomb Potential.

The far-zone onset: asymptopia has a doorstep

" $r\to\infty$ " earns its keep only if we can say how far is far. The answer is the article's owned laboratory.

Go deeper · Derivation C · 2 minThe far-zone onset theorem: the Fraunhofer distance kd²

Derivation CThe far-zone onset theorem: the Fraunhofer distance kd²

We are deriving — then numerically measuring — the radius at which the asymptotic form (3) becomes accurate for a target of size $d$ . The strategy: model the target as two coherent point emitters separated by $d$ (the minimal object with internal structure), expand the exact path lengths, and demand the first dropped term stay below a radian; then, because a derived scale deserves a measurement, compute the exact two-emitter field and locate the knee empirically. Watch the quadratic term — the linear one is not an error at all.

Step 1. Exact and expanded path lengths. An emitter at $\mathbf r'$ ( $|\mathbf r'|\le d/2$ ) reaches the observer at $\mathbf r$ over the distance

|\mathbf r-\mathbf r'| \;=\; r \;-\;\hat{\mathbf r}\cdot\mathbf r' \;+\;\frac{r'^2-(\hat{\mathbf r}\cdot\mathbf r')^2}{2r}\;+\;\cdots

Why the first two terms are kept. The $r$ term is the common spherical phase; the linear term $-\hat{\mathbf r}\cdot\mathbf r'$ , multiplied by $k$ , is precisely the direction-dependent phase $e^{-i\mathbf k'\cdot\mathbf r'}$ that builds $f(\theta,\phi)$ — target structure encoded in angle. Keeping it is what "point target with a directional voice" means.

Step 2. The first dropped phase. The quadratic term contributes a phase error of at most

\Delta\varphi \;\simeq\; \frac{k\,(d/2)^2}{2r}\;=\;\frac{kd^2}{8r}

The asymptotic form is trustworthy when $\Delta\varphi\ll1$ : the far zone begins at $r\sim kd^2$ — larger targets and shorter wavelengths push it outward, quadratically in $d$ .

Step 3. Measure it. Two unit emitters at $z=\pm d/2$ , $kd=10$ ; the asymptotic amplitude is $f(\theta)=2\cos[(kd/2)\cos\theta]$ . Against the exact summed field over $r/d\in[3.2,\,3.2\times10^4]$ : the relative error falls from $0.39$ to $4.0\times10^{-5}$ on a log-log slope of $-1.000$ — the predicted $1/r$ — matching $kd^2/8r$ within a factor of two; the $10\%$ knee lands at $r_{10}=12.9\,d\approx1.3\,kd^2$ . Why slope −1 and not −2. The dropped term is a phase; to first order a phase error $\Delta\varphi$ produces an amplitude error $|e^{i\Delta\varphi}-1|\simeq\Delta\varphi\propto1/r$ . The next-order $1/r^2$ corrections are invisible until the phase error itself is tiny.

Step 4. Reality check. For a thermal neutron on an atomic target, $kd^2\approx3.5\times10^{-10}$ m against a $0.3$ m detector — nine decades inside the far zone; for optical light on a millimetre object, $kd^2\approx13$ m and the far field never arrives on the bench. The full ladder is Worked Example 1. Verified: expansion error slope $-1.000$ measured over the last decade; error/(phase bound) $=1.0$ ; $r_{10}/kd^2=1.29$ ; all ladder numbers recomputed from CODATA constants — 18-check suite green.

Which r matters: the three scales, ordered

scale	meaning	typical size (thermal neutron, atomic target)
$R$ — range of $V$	interior physics; wave function target-specific	$10^{-10}$ m
$kd^2$ — Fraunhofer onset	far zone begins; universal form switches on	$3.5\times10^{-10}$ m
$r_{\mathrm{det}}$ — detector	where (3) is actually used	$0.3$ m

The hierarchy $R\lesssim kd^2\ll r_{\mathrm{det}}$ is what makes scattering theory clean. At $r_{\mathrm{det}}$ the detector-direction card applies: the outgoing wave at $\hat{\mathbf r}$ is locally a plane wave of momentum $\mathbf k'=k\hat{\mathbf r}$ , so placing a counter at $(\theta,\phi)$ is selecting the final momentum — why every formula downstream writes $f(\mathbf k',\mathbf k)$ . One honesty card completes the picture. A real experiment sends wave packets, not infinite stationary beams: a packet of width $\Delta k$ built on the family (3) rides through, scatters, and disperses — construction and spreading laws owned by Free Particle and Wave Packets and Wave Packet Spreading. Averaged over the packet's narrow band, every counting prediction reduces to the stationary state's — which is why the chapter may work at fixed $k$ , and the same band-averaging is what erases the backward interference endpoint in Scattering Amplitude's audit.

Limiting cases

case	result	meaning
$p=1$ falloff	flux through spheres $r$ -independent	the unique conserving power — the theorem
$V\equiv0$	$f=0$ , $\psi=e^{ikz}$ exactly	no target, no ripple; the beam is its own asymptotic form
$r\gg kd^2$	error $\simeq kd^2/8r\to0$	asymptopia; detector nine decades deep for neutron experiments
$r\lesssim kd^2$	no single $f(\theta)$ describes the field	near zone: Fresnel regime, target structure unresolved into one amplitude
$V\sim r^{-n}$ , $n>1$	$\int^\infty V\,dr<\infty$ , form holds	short-range in the scattering sense — even $n=3/2$ qualifies
$V\sim1/r$ (Coulomb)	phase $kr-\eta\ln2kr$ ; form fails	logarithmic distortion, never outrun; screened rescue → Scattering by a Screened Coulomb Potential
$e^{-ikr}/r$ selected instead	$j_r<0$ : convergent conspiracy	the time-reversed solution; legitimate math, wrong experiment

Worked examples

Example 1 — the Fraunhofer ladder for real beams. For (a) a thermal neutron ( $E=25$ meV) on an atom ( $d=1$ Å), (b) a $10$ keV X-ray on the same atom, and (c) the same neutron on a Rydberg atom with $n=50$ : find the far-zone onset $kd^2$ and compare with a detector at $r=0.3$ m.

Step 1. Neutron wavenumber. $p=\sqrt{2m_nE}$ with $E=25$ meV gives $\lambda=h/p=1.81$ Å and $k=2\pi/\lambda=3.47\times10^{10}\,\mathrm{m^{-1}}$ .

Step 2. Onset for the atomic target: $kd^2=(3.47\times10^{10})(10^{-10})^2=\mathbf{3.5\times10^{-10}\;m}$ . The detector ratio is $0.3/3.5\times10^{-10}=\mathbf{8.6\times10^{8}}$ — nine orders of magnitude inside the far zone.

Step 3. X-ray: $\lambda=1.24$ Å $\Rightarrow k=5.07\times10^{10}\,\mathrm{m^{-1}}$ , so $kd^2=\mathbf{5.1\times10^{-10}\;m}$ — same conclusion; wavelength parity with the neutron keeps the scale atomic.

Step 4. Rydberg target: $d=n^2a_0=0.13\,\mu$ m, so $kd^2=\mathbf{0.61\;mm}$ — a million times the atomic value (the $d^2$ scaling has teeth), yet still inside a $0.3$ m apparatus. Answer: $3.5\times10^{-10}$ m, $5.1\times10^{-10}$ m, $0.61$ mm — all $\ll0.3$ m; the asymptotic form is practically exact in all three experiments.

Example 2 — reading the measured error curve. For the two-emitter laboratory with $kd=10$ : how accurate is the asymptotic form at $r=3.2\,d$ , at the knee, and at $r=3.2\times10^4\,d$ — and what does the phase bound predict at the far point?

Step 1. Near point: at $r=3.2\,d$ the measured error is $\mathbf{0.39}$ — off by forty percent three target-lengths out. The near zone is real.

Step 2. The knee: the error crosses $10\%$ at $r_{10}=\mathbf{12.9\,d}$ , i.e. $1.29\,kd^2$ in Fraunhofer units — the derived scale, located empirically within thirty percent.

Step 3. Far point: at $r=3.2\times10^4\,d$ the measured error is $\mathbf{4.0\times10^{-5}}$ . The quadratic-phase bound predicts $kd^2/8r=10\times d/(8\times3.2\times10^4 d)=3.9\times10^{-5}$ — agreement to a few percent, and the interpolating slope is $-1.000$ : pure $1/r$ , as the dropped-phase argument demands. Answer: 39% → 10% at $12.9\,d$ → $4\times10^{-5}$ , falling exactly as $kd^2/8r$ .

Origins · 1 minFar Field and how this was discovered

Far Field

1818–1912 · Paris, Munich

The far field was charted long before quantum mechanics needed it. Fresnel's 1818 prize memoir built the near-zone theory — his integrals and half-period zones describe exactly the region where a source's parts still argue by phase. Fraunhofer, working with gratings in Munich from 1821, isolated the opposite regime: far enough away (or behind a lens) the pattern depends only on direction. Kirchhoff made the split mathematical in 1882 with his diffraction integral, and in 1912 Sommerfeld supplied the uniqueness clause — the radiation condition, requiring outgoing behaviour at infinity, without which the exterior problem has spurious solutions (his person-story is told elsewhere in this program; here he contributes one boundary condition). Quantum scattering inherited the toolkit wholesale: the Fraunhofer distance $kd^2$ , the outgoing condition, the direction-reads-momentum card — hundred-year-old optics, re-employed at $10^{-10}$ m.

History

1818

— Fresnel submits his memoir on diffraction, crowned by the Académie the next year: the near zone gets its integrals

1821

— Fraunhofer's grating work isolates the direction-only regime that now bears his name

1882

— Kirchhoff derives the diffraction integral, formalizing the near/far split

1909

— Debye's rigorous wave asymptotics sharpen the large-kr toolkit

1912

— Sommerfeld states the radiation condition: outgoing at infinity makes exterior problems unique

1950

— Lippmann and Schwinger embed the outgoing condition into the +iε of formal theory

Going further

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., Ch. 11 (Scattering), Sec. 11.1-11.2 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 6 (Scattering Theory) · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 19 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. VIII · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 13 · Landau & Lifshitz, Quantum Mechanics (Non-relativistic Theory), Vol. 3, Ch. XVII

Provenance. All equations are standard textbook results. A verifier should: (1) confirm the asymptotic form and $\mathrm{d}\sigma/\mathrm{d}\Omega=|f|^2$ against Griffiths 3rd ed. Sec. 11.1-11.2 and Sakurai 2nd ed. Sec. 6.1; (2) check the Green's function $G^{(+)}=-(1/4\pi)e^{ik|\mathbf{r}-\mathbf{r}'|}/|\mathbf{r}-\mathbf{r}'|$ and the amplitude integral $f=-(m/2\pi\hbar^2)\int e^{-i\mathbf{k}'\cdot\mathbf{r}'}V\psi^{(+)}\mathrm{d}^3r'$ against Sakurai Sec. 6.1 or Shankar Ch. 19; (3) verify the Rayleigh expansion and asymptotic $j_\ell(kr)\to\sin(kr-\ell\pi/2)/kr$ (Abramowitz & Stegun / Arfken); (4) verify the momentum transfer relation $|\mathbf{q}|=2k\sin(\theta/2)$ and the optical theorem $\sigma_{tot}=(4\pi/k)\mathrm{Im}\,f(0)$ (Cohen-Tannoudji Vol. II Ch. VIII). The Sommerfeld parameter $\eta=Z_1Z_2 e^2 m/\hbar^2 k$ and the Coulomb logarithmic phase appear in Landau-Lifshitz Vol. 3 Ch. XVII. No experimental numbers are quoted; only structural and symbolic results. (Lane note: of the base's menu this article keeps and re-verifies the asymptotic-form and 1/r-flux derivations; the base's $|f|^2$ steps belong to Scattering Amplitude, its Rayleigh-split derivation to the canon owner Partial Wave Expansion of Plane Waves, its Green's-function/amplitude-integral blocks to Lippmann-Schwinger Equation and Green's Function for Scattering, its optical-theorem derivation to Optical Theorem, and its Born paragraph to Born Approximation — relocated by chapter ownership; no physics error was found in the base.)

Builds on: Time-Independent Schrödinger Equation · Free Particle and Plane Waves · Probability Current and the Continuity Equation · Spherical Coordinates and the Helmholtz Equation Leads to: Scattering Amplitude · Differential and Total Cross Section · Lippmann-Schwinger Equation · Green's Function for Scattering · Partial-Wave Expansion · Optical Theorem

If the plane wave extends over all space, how can the state describe a localized experiment?

It describes the experiment's steady state. A real run sends wave packets — finite trains built from a narrow band of these stationary states (owners: Free Particle and Wave Packets, Wave Packet Spreading) — and after band-averaging, every counting rate reduces to the fixed- $k$ prediction. The stationary state is to the beam what a DC solution is to a circuit: transients have simply left the observation window.

Is e^(ikr)/r actually a solution of the Schrödinger equation, or only approximately?

Exactly, for every $r>0$ — substitution into $(\nabla^2+k^2)$ gives zero identically. The approximation is never about the wave, only the superposition: near the target you need many such waves with different centres to represent the field; beyond $kd^2$ a single centred one suffices. At $r=0$ the function is singular and $(\nabla^2+k^2)[e^{ikr}/4\pi r]=-\delta^3(\mathbf r)$ — the defining property of the kernel Green's Function for Scattering owns.

The time-independent equation is invariant under complex conjugation — so why isn't the incoming solution equally physical?

It is equally mathematical — the time reverse of the outgoing state, and formal theory uses it (the ψ⁻ family, paired with $E-i\epsilon$ ; owners: Retarded and Advanced Green Functions, Lippmann-Schwinger Equation). It fails only as a description of this experiment: its current converges on the target, requiring the walls to emit a perfectly phased spherical conspiracy. The boundary condition is where the experiment's arrow enters a time-symmetric formalism.

Does "short-range" mean the potential has to vanish beyond some radius?

No — that would exclude nearly everything real. The criterion is convergence of $\int^\infty V\,dr$ : any $r^{-n}$ with $n>1$ qualifies, Yukawa exponentials trivially. The boundary case $n=1$ (Coulomb) fails logarithmically — measurably, through the $\eta\ln2kr$ phase. In between sit long-range curiosities (dipole $1/r^2$ tails) where the form holds but partial-wave sums converge slowly — flagged here, handled downstream.

How can the detector direction "measure" momentum when the wave spreads over all angles?

The outgoing wave carries all final momenta at once — one amplitude per direction, which is what $f(\theta,\phi)$ tabulates. A detector at $\hat{\mathbf r}$ interacts with the locally-plane wavefronts crossing it (wavevector $k\hat{\mathbf r}$ ); a count collapses the superposition onto that momentum. At $r\gg kd^2$ the correspondence is clean to within the dropped $kd^2/8r$ phase.

Why do optics and quantum scattering use the same Fraunhofer criterion?

Both solve the Helmholtz equation with a finite source, and $kd^2$ is a property of the equation, not the field's interpretation. Fresnel and Fraunhofer charted the zones for light a century early; the probability amplitude inherited the geometry unchanged. The quantum addition is the reading: here the far field carries $|f|^2$ — a cross section, with Scattering Amplitude's current theorem converting geometry into counting rates.