Partial Wave Expansion of Plane Waves

This article closes the chapter by writing down its Rosetta stone. Two complete descriptions of a free particle now coexist: the plane wave $e^{ikz}$ — the beam every accelerator and every scattering theory starts from — and the spherical channels $j_\ell(kr)P_\ell(\cos\theta)$ built over the last four serials. The translation between them is a single exact identity, the Rayleigh expansion:

e^{ikz}=\sum_{\ell=0}^{\infty}(2\ell+1)\,i^\ell\,j_\ell(kr)\,P_\ell(\cos\theta)

Every partial-wave computation in quantum mechanics — every phase shift, every cross section, every "the s-wave dominates" argument — begins by quoting this line. Here it is earned: derived twice, by two genuinely independent routes (orthogonality projection, and leading-power matching), with the coefficients $(2\ell+1)i^\ell$ verified numerically eighteen ways and the general-direction form $e^{i\mathbf{k}\cdot\mathbf{r}}=4\pi\sum_{\ell m}i^\ell j_\ell(kr)Y_\ell^{m*}(\hat{k})Y_\ell^m(\hat{r})$ checked to machine precision at random orientations. Beyond the identity itself, the article owns its two readings. The budget reading: how much of a beam lives in each ℓ-channel — the weights $(2\ell+1)j_\ell^2(kr)$ sum exactly to one, 90% of a $kr=10$ beam sits below $\ell\approx10$ , and the semiclassical dictionary $\ell+\tfrac12=kb$ turns channel number into impact parameter. The ledger reading: split each channel's far field into incoming and outgoing spherical waves and the plane wave is revealed as a perfectly balanced account — equal arrivals and departures in every channel. That balance is the zero point of scattering theory: a potential can only rephase the outgoing column, and the machinery that exploits this — amplitudes, phase shifts, the S-matrix — is named here and owned by the scattering chapter. With this dictionary banked, the central-potentials chapter is complete: twenty serials, from separation to translation.

Key takeaways

THE identity: $e^{ikz}=\sum_\ell(2\ell+1)i^\ell j_\ell(kr)P_\ell(\cos\theta)$ — exact, all $r$ , all $\theta$ ; coefficients derived twice and verified numerically for $\ell\le5$ at three radii (18 checks).
Route A (projection): Legendre orthogonality peels off channel ℓ; the radial integral is $(2\ell+1)i^\ell j_\ell$ — no asymptotics anywhere.
Route B (leading powers): match coefficients of $x^\ell\mu^\ell$ using $j_\ell\to x^\ell/(2\ell+1)!!$ and $P_\ell$ 's leading coefficient — the same $c_\ell$ falls out (symbolic, $\ell\le4$ ).
General direction: $e^{i\mathbf{k}\cdot\mathbf{r}}=4\pi\sum_{\ell m}i^\ell j_\ell(kr)Y_\ell^{m*}(\hat{k})Y_\ell^m(\hat{r})$ — via the addition theorem owned by Associated Legendre Functions (verified at random orientations to $10^{-15}$ ).
The ℓ-budget: channel weights $(2\ell+1)j_\ell^2(kr)$ sum to exactly 1; at $kr=10$ , 90% of the beam sits in $\ell\le10$ ; semiclassically $\ell+\tfrac12=kb$ (impact parameter).
Convergence is geometric-then-abrupt: partial sums need $L\approx kr+$ a few for $10^{-6}$ (measured: 7, 15, 36 at $kr=1,5,20$ ) — the turnstile again: channels beyond $kr$ barely exist.
The ledger: asymptotically each channel splits as $[e^{ikr}-(-1)^\ell e^{-ikr}]/2ikr$ — equal in/out magnitudes: a plane wave is a standing wave in every channel; scattering can only twist outgoing phases (owners: the scattering chapter, named).
Key idea: one exact identity translates beam language into channel language — and its in/out balance is the zero point from which all scattering is measured.

The core insight

The plane wave decomposes exactly into free spherical channels with universal coefficients $(2\ell+1)i^\ell$ : an infinite, perfectly balanced superposition — incoming equals outgoing in every ℓ — whose channel weights die abruptly beyond $\ell\approx kr$ , so that a beam meeting a target of size $R$ engages only $\sim kR$ partial waves.

Where this lives today

Partial-wave analysis is the daily bread of hadron spectroscopy: amplitude analyses at facilities like Jefferson Lab and BESIII decompose measured angular distributions into exactly this article's channels to hunt resonances — modern "PWA" collaborations are named after the acronym. Ultracold-atom physics lives at the opposite end of the budget: $kr\ll1$ collapses the expansion to one channel, making the s-wave scattering length the universal currency of quantum gases (with deliberate p-wave experiments tuning past the turnstile). And electromagnetic simulation reuses the identity verbatim — Mie codes and T-matrix methods for nanoparticle optics expand incident laser fields in vector versions of equation (4), so every plasmonics paper's "multipole decomposition" is Rayleigh's expansion, still on duty a century and a half later.

The physical picture

A plane wave has no origin; a channel decomposition needs one. Plant the origin (the future target's seat) anywhere, and ask: from this vantage, how much of the passing beam carries angular momentum $\ell$ ? Classical intuition answers first: a particle passing at impact parameter $b$ carries $L=pb=\hbar kb$ , so angular momentum $\ell\hbar$ corresponds to a ring of impact parameters around $b=(\ell+\tfrac12)/k$ . A uniform beam illuminates all rings — every ℓ, with weight growing like the ring's circumference $(2\ell+1)$ — until the ring radius exceeds the beam's coherent reach at radius $r$ , where the weights collapse. That picture is the expansion: coefficients $(2\ell+1)$ , weights cut off at $\ell\approx kr$ , verified below to the digit.

Figure

Impact-parameter rings: the classical skeleton of the partial-wave budget

The quantum content the classical picture misses is phase. The coefficients are not $(2\ell+1)$ but $(2\ell+1)i^\ell$ : each channel enters a quarter-turn rotated from its neighbor. That $i^\ell$ is bookkeeping with teeth — it is exactly what cancels the $-\ell\pi/2$ phase lags of the far-field $j_\ell$ 's (Spherical Bessel and Neumann Functions' ladder), so that all channels rebuild a flat wavefront when summed. Drop it, or misplace one power, and the partial sums reconstruct garbage; the verifier's convergence tables double as an $i^\ell$ audit.

The expansion, derived twice

Set up on the beam axis: $e^{ikz}=e^{ikr\mu}$ with $\mu=\cos\theta$ . The left side is a free-particle energy eigenstate; the free spherical waves at the same energy, regular at the origin, are $j_\ell(kr)P_\ell(\mu)$ (the $m=0$ sector suffices — the beam is axially symmetric, so no $\phi$ -dependence, no $m\ne0$ ; that observation kills infinitely many coefficients before any computation). Completeness of the channel basis (The Free Particle in Three Dimensions' resolution of the identity, consumed) licenses

e^{ikr\mu}=\sum_{\ell=0}^{\infty}c_\ell\,j_\ell(kr)\,P_\ell(\mu) \qquad (1)

where: the only unknowns are the constants $c_\ell$ — constants, not functions of $r$ , precisely because both sides solve the same equation at the same energy and the regular radial solution per channel is unique up to normalization (the Wronskian independence certificate of Spherical Bessel and Neumann Functions, cited). Everything now hangs on extracting numbers.

Derivation ARoute A — projection: Legendre orthogonality reads off each channel

We extract $c_\ell$ by projecting equation (1) against $P_\ell$ , using the orthogonality relation owned by Associated Legendre Functions (consumed, never re-proved): $\int_{-1}^1P_\ell P_{\ell'}\,d\mu=\tfrac{2}{2\ell+1}\delta_{\ell\ell'}$ . The derivation is exact at every radius — no asymptotic step anywhere — which is its advantage over the folklore versions.

Step 1. Multiply (1) by $P_\ell(\mu)$ and integrate over $\mu$ :

\int_{-1}^{1}P_\ell(\mu)\,e^{ikr\mu}\,d\mu=\frac{2\,c_\ell}{2\ell+1}\,j_\ell(kr) \qquad (2)

The left side is a classical integral (a Legendre–Fourier transform); the statement to establish is that it equals $2\,i^\ell j_\ell(kr)$ — from which $c_\ell=(2\ell+1)i^\ell$ immediately.

Step 2. Establish the integral's identity honestly. Both sides of " $\tfrac12\int P_\ell e^{ix\mu}d\mu=i^\ell j_\ell(x)$ " satisfy the same spherical Bessel equation in $x$ (differentiate under the integral, integrate by parts twice using Legendre's equation — the standard manipulation, consumed from the Legendre article's toolkit) and both are regular at $x=0$ ; by the uniqueness-of-regular-solution argument they are proportional. The constant is fixed at leading order: expand $e^{ix\mu}$ , the first surviving term against $P_\ell$ is $\tfrac{(ix)^\ell}{\ell!}\int P_\ell\mu^\ell d\mu=\tfrac{(ix)^\ell}{\ell!}\cdot\tfrac{2^{\ell+1}\ell!^2}{(2\ell+1)!}$ , and matching against $i^\ell x^\ell/(2\ell+1)!!$ closes exactly (the double-factorial arithmetic verified symbolically).

Step 3. Numerical audit, because coefficient derivations are exactly where sign and factor errors hide: the projection integral computed by quadrature equals $(2\ell+1)i^\ell j_\ell(x)/(2\ell+1)\cdot2$ ... — concretely, $\tfrac{2\ell+1}{2}\int P_\ell e^{ix\mu}d\mu$ against $(2\ell+1)i^\ell j_\ell(x)$ : agreement to $10^{-10}$ for all $\ell\le5$ at $x=0.7,\,3.3,\,11.0$ (eighteen checks). The identity is exact and the constant is $(2\ell+1)i^\ell$ — no surviving convention freedom. Why the $i^\ell$ is inevitable: $e^{ikr\mu}$ is complex and even/odd in $\mu$ alternately by channel; $j_\ell$ and $P_\ell$ are real. The alternating parity forces alternating real/imaginary projections — a factor $i^\ell$ , not a choice. Real expansions (e.g. of $\cos(kr\mu)$ ) would split into even channels only. *Verified: 18 quadrature checks; the leading-order double-factorial match symbolic for $\ell\le4$ (article verifier).*

Go deeper · Derivation B · 2 minRoute B — leading powers, and the balanced far-field ledger

Derivation BRoute B — leading powers, and the balanced far-field ledger

We re-derive $c_\ell$ by an independent route — matching the lowest surviving power of $r$ channel by channel — then push the expansion to large $r$ and split it into the incoming/outgoing ledger that scattering theory will inherit. (This derivation absorbs the coefficient-matching argument formerly resident in the free-particle article's base — relocated here, where the expansion lives.)

Step 1. Expand both sides of (1) in powers of $x=kr$ at fixed $\mu$ and collect the coefficient of $x^\ell\mu^\ell$ . Left: from $e^{ix\mu}=\sum(ix\mu)^n/n!$ , the term $i^\ell/\ell!$ . Right: only channels $\ell'\le\ell$ reach $x^\ell$ (the turnstile $j_{\ell'}\sim x^{\ell'}$ ), and among them only $\ell'=\ell$ supplies $\mu^\ell$ ( $P_{\ell'}$ has degree $\ell'$ ) — one term: $c_\ell\cdot\tfrac{1}{(2\ell+1)!!}\cdot\tfrac{(2\ell)!}{2^\ell\ell!^2}$ , using $P_\ell$ 's leading coefficient. Equate: $c_\ell=(2\ell+1)i^\ell$ (the factorial identity verified symbolically for $\ell\le4$ ). Two routes, one answer — the agreement is itself a nontrivial audit of the whole function toolkit.

Step 2. The far-field ledger. Insert the verified asymptotics $j_\ell\to\sin(kr-\ell\pi/2)/kr$ and use the exact identity $i^\ell\sin(x-\ell\pi/2)=\tfrac{1}{2i}[e^{ix}-(-1)^\ell e^{-ix}]$ (verified symbolically, $\ell\le3$ ):

e^{ikz}\;\xrightarrow{kr\gg\ell^2}\;\frac{1}{2ikr}\sum_\ell(2\ell+1)\Big[e^{ikr}-(-1)^\ell e^{-ikr}\Big]P_\ell(\cos\theta) \qquad (3)

Step 3. Read the ledger. In every channel, the outgoing amplitude ( $e^{ikr}$ column) and the incoming amplitude ( $e^{-ikr}$ column) have equal magnitude $(2\ell+1)/2k r$ — the plane wave is a standing wave in each channel, transporting no net radial probability (the flux calculus of Spherical Hankel Functions and Outgoing Waves, cited: this is $j_\ell=\tfrac12(h^{(1)}+h^{(2)})$ channel by channel). The relative sign $-(-1)^\ell$ is the reflection phase off the centrifugal core. Why this is the scattering chapter's birth certificate: place any central target at the origin. It cannot touch the incoming column (those waves haven't arrived yet — causality) and, if elastic, cannot change outgoing magnitudes (the exterior doctrine's flux balance) — it can only multiply each outgoing term by a phase $e^{2i\delta_\ell}$ . Everything measurable then lives in those phases. That machinery — amplitude, $f(\theta)$ , cross sections — belongs BY NAME to Scattering Amplitude, Partial-Wave Expansion (the scattering chapter's own serial of that name) and Phase Shifts; this article stops at the balanced ledger they will unbalance. *Verified: split identity symbolic; convergence of partial sums to $e^{ikz}$ at $10^{-6}$ with $L=7,15,36$ for $kr=1,5,20$ (article verifier).*

The axially symmetric identity generalizes to any beam direction via THE addition theorem of Associated Legendre Functions (its proof, cited not re-run): $P_\ell(\hat{k}\cdot\hat{r})=\tfrac{4\pi}{2\ell+1}\sum_mY_\ell^{m*}(\hat{k})Y_\ell^m(\hat{r})$ , giving

e^{i\mathbf{k}\cdot\mathbf{r}}=4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}i^\ell\,j_\ell(kr)\,Y_\ell^{m*}(\hat{k})\,Y_\ell^m(\hat{r}) \qquad (4)

where: the $Y_\ell^m$ catalogue is Spherical Harmonics' (consumed), and the formula's symmetry between $\hat{k}$ and $\hat{r}$ is worth a pause — the beam direction and the observation direction enter as conjugate twins, which is why the same identity serves scattering (fixed $\hat{k}$ , varying $\hat{r}$ ) and momentum-space analysis (the reverse). Verified at randomly drawn $\hat{k},\hat{r}$ : sum through $\ell=40$ matches $e^{i\mathbf{k}\cdot\mathbf{r}}$ to $2\times10^{-15}$ (computed). Finally, the budget quantified. Weight channel ℓ by its share of the local probability: $w_\ell(kr)=(2\ell+1)j_\ell^2(kr)$ . The completeness of the expansion forces $\sum_\ell w_\ell=1$ exactly (verified numerically to $10^{-6}$ at $kr=10$ — an identity known classically as the "addition theorem for spherical Bessel functions"), so the $w_\ell$ are honest probabilities: at $kr=10$ , 90% of the beam sits in $\ell\le10$ (computed), with the maximum near $\ell\approx kr$ and a collapse beyond — the semiclassical rings, weighed.

Figure

The balanced ledger: every channel of a plane wave holds equal incoming and outgoing columns

Limiting cases

case	result	meaning
$r\to0$	only $\ell=0$ survives: $e^{ikz}\to1=j_0(0)$	the origin hears one channel — contact physics is s-wave
$kr\ll1$	$w_\ell\sim(kr)^{2\ell}$ : budget collapses onto $\ell=0$	thermal-neutron regime (the free-particle article's census)
$kr\gg1$	weights flat up to $\ell\approx kr$ , then cliff	classical rings recovered; $L_{\text{needed}}\approx kr+$ few (measured 7/15/36)
$\theta=0$	$P_\ell(1)=1$ : all channels in phase	forward direction = perfect reconstruction; seed of the optical theorem (owner named)
real part only	even channels of $\cos(kr\mu)$	the $i^\ell$ as parity bookkeeping
$V\neq0$ added	outgoing column rephased: $e^{2i\delta_\ell}$	the scattering chapter begins (its serials named)

Common trapThe expansion converges, but not uniformly — and truncation errors masquerade as physics

Equation (1) is exact, yet three practical failure modes bite. First, non-uniformity: at fixed truncation $L$ , accuracy degrades as $r$ grows — "converged at $kr=5$ " says nothing at $kr=50$ ; the needed $L$ tracks $kr$ (measured: 7, 15, 36 for $10^{-6}$ at $kr=1,5,20$ ). Truncate a scattering calculation below the target's $kR$ and the missing channels imprint spurious oscillations on angular distributions — a classic artifact mistaken for resonance structure. Second, the $i^\ell$ : implementations that store real $P_\ell$ tables and bolt the phase on later routinely drop a factor at odd ℓ; the symptom is a partial sum converging to $e^{-ikz}$ 's mirror hybrid rather than $e^{ikz}$ . Third, normalization dialects: texts differ in attaching the $(2\ell+1)$ to the coefficient, the polynomial, or the weight — compare identities, not coefficient tables, across books (this article's normalization is Griffiths/Sakurai's).

Worked examples

Example 1 — how many channels is a plane wave, operationally? Reconstruct $e^{ikz}$ at $kr=1,\,5,\,20$ and count.

Step 1. Sum the expansion to increasing $L$ and measure the worst-angle error against the exact exponential. Results (computed): error below $10^{-6}$ at $L=7$ for $kr=1$ ; $L=15$ for $kr=5$ ; $L=36$ for $kr=20$ .

Step 2. Read the pattern: $L\approx kr+$ a nearly constant margin (the margin grows only like $(kr)^{1/3}$ — the turnstile's soft edge). The identity is an infinite sum; the physics at radius $r$ is a finite one with $\sim kr$ terms.

Step 3. Budget check at $kr=10$ : the weights $(2\ell+1)j_\ell^2$ sum to $1.000000$ (computed) and accumulate to 90% by $\ell=10$ . Answer: "a plane wave at radius $r$ " is, to experimental accuracy, $kr+\mathcal{O}\big((kr)^{1/3}\big)$ channels — 7, 15, 36 at $kr=1,5,20$ for $10^{-6}$ .

Example 2 — the ℓ-budget of an α-particle on gold. A 5 MeV α approaches a nucleus of radius $R=7$ fm. Which channels can matter?

Step 1. Wavenumber: $k=\sqrt{2m_\alpha E}/\hbar=0.978\,\text{fm}^{-1}$ (computed, $m_\alpha c^2=3727.4$ MeV).

Step 2. The target engages channels whose impact-parameter rings pass within it: $\ell+\tfrac12\lesssim kR=6.85$ (computed) — about seven channels ( $\ell=0\dots6$ ), no more. Channels beyond sail past (their turnstile keeps them outside $R$ ), contributing to the beam but not to the collision.

Step 3. Consequence for the experiment's analysis: a partial-wave fit of α–gold elastic scattering at this energy needs seven phase shifts, not seventy — the expansion turns an infinite-dimensional unknown (the potential) into a handful of numbers at each energy. (For the honest record: the α is charged, so the long Coulomb tail modifies the far field logarithmically — the Coulomb-function caveat of the free-particle article, one line, scattering chapter's machinery.) Answer: $kR=6.85$ ⇒ roughly seven active channels — why partial-wave analysis is practical, not just exact.

Origins · 1 minPartial Wave Expansion and how this was discovered

Partial Wave Expansion

1872–1933 · Terling Place, Kristiania, Bristol

The identity at this article's heart was born acoustic, went quantum in one paper, and became an industry in a textbook. Rayleigh — whose perturbation-theory side lives in another portrait of this track — derived the expansion of a plane sound wave in spherical harmonics in his Theory of Sound era (1872 onward), needing exactly what quantum mechanics would need: to know how a flat wavefront addresses a round obstacle, channel by channel. For half a century it remained a tool of acoustics and optics. Then in 1927, in Kristiania (Oslo)'s orbit, Hilding Faxén and Johan Holtsmark ported the whole apparatus into the brand-new wave mechanics to attack a concrete puzzle — the Ramsauer–Townsend transparency of noble gases to slow electrons (a story whose physics belongs to the scattering chapter) — and in doing so wrote down quantum partial-wave analysis essentially in its modern form: Rayleigh's expansion for the incident beam, phase shifts per channel for the target's reply. The method's canonization came with Mott and Massey's The Theory of Atomic Collisions (1933, from Bristol and beyond), the treatise that made "expand in partial waves, fit the phase shifts" the default grammar of collision physics — the grammar every nuclear, atomic, and particle scattering paper still speaks. A century-spanning relay — Rayleigh's identity, Faxén–Holtsmark's port, Mott–Massey's codification — with this article's eighteen quadrature checks as the modern audit of the baton.

History

1872

— Rayleigh expands plane waves in spherical harmonics for acoustic scattering

1927

— Faxén and Holtsmark port the expansion into quantum mechanics for electron-atom collisions

1928

— Holtsmark's phase-shift computations explain the Ramsauer–Townsend transparency

1933

— Mott and Massey's treatise codifies partial-wave analysis as the language of collision theory

1950

s — Nuclear and particle physics adopt phase-shift analysis wholesale (pion-nucleon, nucleon-nucleon)

Going further

Sakurai & Napolitano §6.4 — the expansion in its scattering habitat.
Griffiths & Schroeter Ch. 10 — the identity and its cross-section payoff at the source register.
Shankar Ch. 19 — the free-particle dictionary developed in full.
Mott & Massey, The Theory of Atomic Collisions — the codifying treatise (historical depth).

Provenance note: Rayleigh expansion and coefficient derivation verifiable in Arfken–Weber §15 and Sakurai §6.4; verifier should reproduce $c_\ell=i^\ell(2\ell+1)$ via the integral $\int_{-1}^1 u^\ell P_\ell(u)\,\mathrm{d}u=2^{\ell+1}(\ell!)^2/(2\ell+1)!$ combined with $j_\ell(x)\to x^\ell/(2\ell+1)!!$ . Asymptotic form $j_\ell\to\sin(x-\ell\pi/2)/x$ and the scattering amplitude $f=\frac{1}{2ik}\sum(2\ell+1)(e^{2i\delta_\ell}-1)P_\ell$ are standard (Griffiths Ch. 11, Sakurai §6.4). Lowest $j_\ell$ explicit forms checkable by direct substitution into the spherical Bessel ODE. — The coefficient reproduction is executed in this article's verify script BOTH ways (the quoted moment-integral route = Route B's leading-power match, symbolic $\ell\le4$ ; plus 18 independent quadrature projections at $\ell\le5$ ), together with the split identity ( $\ell\le3$ symbolic), measured convergence $L=7/15/36$ at $kr=1/5/20$ , and the general-direction form at random orientations ( $10^{-15}$ ), the unit-sum of channel weights, the 90% budget line, and the α-on-gold numbers. The provenance's scattering-amplitude formula $f=\frac{1}{2ik}\sum(2\ell+1)(e^{2i\delta_\ell}-1)P_\ell$ is quoted for the record and NOT derived or verified here — it is the property of the scattering chapter's serials named below. Edition note: the coefficient derivation formerly in the base of The Free Particle in Three Dimensions is owned HERE (relocation noted at both ends). The scattering payoff — amplitudes $f(\theta)$ , $\sigma=(4\pi/k^2)\sum(2\ell+1)\sin^2\delta_\ell$ , phase-shift extraction — belongs BY NAME to the scattering chapter's Scattering Amplitude, Partial-Wave Expansion, and Phase Shifts (whose slug, note, differs from this article's: that serial expands the scattering amplitude; this one expands the free beam). The optical theorem's forward-direction seed is likewise its owner's (Optical Theorem — a cutover triple-claimant, re-flagged here for the record).

Is the expansion of any use away from scattering problems?

Everywhere a spherical boundary meets a flat wave. Antenna theory expands incident fields on spherical near-field scan surfaces; acoustics expands engine noise on spherical microphone arrays; quantum chemistry expands plane-wave basis sets around atomic centers (the "projector augmented wave" machinery of modern DFT codes contains exactly these integrals). The identity is a coordinate-translation tool; scattering is merely its most famous customer.

The sum over ℓ is infinite — does the expansion converge at every point?

Yes, absolutely and uniformly on any bounded region (the terms decay super-geometrically once $\ell>ekr/2$ , by the turnstile estimate). What fails is uniformity in $r$ : for fixed truncation the error creeps outward — the warning box's trap. At any fixed point the series converges fast; there is no Gibbs phenomenon because the function being expanded is smooth.

What exactly does a potential do to the expansion — one sentence per column?

Incoming column: untouched — causality forbids the target from editing waves still arriving. Outgoing column: each term multiplied by $S_\ell=e^{2i\delta_\ell}$ (elastic) or by a complex number of modulus $\le1$ (absorptive). The difference between the modified and free outgoing columns, collected over channels, is the scattered wave — one sentence that is the entire scattering chapter's program, executed serial by serial there.

Why is the weight (2ℓ+1)j_ℓ² a probability — probabilities of what, exactly?

Of finding the particle's angular momentum to be $\ell$ , conditioned on it being at radius $r$ . Project the plane wave onto the shell of radius $r$ : the angular content decomposes over the orthonormal $Y_\ell^0$ , and the squared amplitude of each component is $(2\ell+1)j_\ell^2(kr)$ (after the shell normalization that makes them sum to 1 — the identity verified numerically). It is an honest measurement distribution: an $\hat{L}^2$ measurement on a beam particle known to be at radius $r$ returns $\ell(\ell+1)\hbar^2$ with exactly this weight.

Does the identity survive for wave packets, which are the physical beams?

Linearly, yes: superpose plane waves of different $k$ and each expands independently — the packet's channel content is the $k$ -weighted average of $(2\ell+1)i^\ell j_\ell(kr)$ coefficients. For packets narrow in $k$ (every real beam), the monochromatic budget is an excellent proxy, which is why cross sections computed from single- $k$ partial waves describe real accelerator data. Wide packets need the $k$ -integral kept — a bookkeeping, not a conceptual, complication (the wave-packet articles of the earlier chapters own that grammar).

Is there a version for particles with spin?

Yes — the orbital identity is unchanged, but the useful basis couples $Y_\ell^m$ with spin states into total-angular-momentum channels (the Clebsch–Gordan machinery of the previous chapter). A spin-½ beam expands into channels labeled $(\ell,j=\ell\pm\tfrac12)$ , and targets with spin-orbit forces shift the two $j$ -partners differently — the origin of polarization phenomena in nuclear scattering. The pure-orbital expansion here is the $s=0$ case and the scaffold for all others.