Spherical Hankel Functions and Outgoing Waves

The previous article built the standing-wave basis: $j_\ell$ and $n_\ell$ , real functions rippling in place. But physics at a target does not stand still — probability arrives and probability leaves, and the natural basis for saying so is not sines and cosines but complex exponentials. This article performs the change of basis: the spherical Hankel functions $h_\ell^{(1)}=j_\ell+in_\ell$ and $h_\ell^{(2)}=j_\ell-in_\ell$ , the travelling-wave alphabet of three-dimensional quantum mechanics. The far field tells you instantly which is which: $h_\ell^{(1)}\to(-i)^{\ell+1}e^{ikr}/kr$ , and once the time factor $e^{-iEt/\hbar}$ is attached, $e^{i(kr-\omega t)}$ moves outward — $h^{(1)}$ radiates, $h^{(2)}$ converges. The probability current makes the label quantitative: an $h_0^{(1)}$ wave carries a computed, $r$ -independent outward flux — probability streaming through every sphere at the same rate, the spherical version of a plane wave's steady current. That outgoing/incoming split is the boundary-condition language of every problem where something leaves: scattered waves, decaying states, radiation. Two payoffs anchor the article. First, the exterior doctrine: outside any finite potential region, the radial solution must be a combination $\alpha h^{(1)}+\beta h^{(2)}$ per channel, and flux bookkeeping constrains the mix — the stage on which the scattering chapter's S-matrix will stand (stated here with its owners named; the matching machinery itself is theirs). Second, the evanescent continuation: substitute the bound-state wavenumber $k\to i\kappa$ and the outgoing wave collapses, exactly, to a decaying exponential — $h_0^{(1)}(i\kappa r)=-e^{-\kappa r}/\kappa r$ , verified symbolically. Bound-state tails are Hankel functions of imaginary argument, which is why the same function that describes a radiated wave also describes the deuteron's halo ( $1/\kappa=4.32$ fm, computed) — and why its $\ell=0$ form is, to the letter, the Yukawa function of the portrait's story.

Key takeaways

Definitions: $h_\ell^{(1)}=j_\ell+in_\ell$ (outgoing), $h_\ell^{(2)}=j_\ell-in_\ell$ (incoming); closed forms $h_0^{(1)}=-ie^{ix}/x$ , $h_1^{(1)}=-(1/x+i/x^2)e^{ix}$ , $h_2^{(1)}=(i/x-3/x^2-3i/x^3)e^{ix}$ — all verified symbolically.
Far field: $h_\ell^{(1)}\to(-i)^{\ell+1}e^{ix}/x=-ie^{i(x-\ell\pi/2)}/x$ — a pure outgoing exponential with the familiar quarter-turn ladder (prefactors verified for $\ell\le2$ ).
Current certificate: $\psi=A\,h_0^{(1)}(kr)Y_0^0$ carries total outward flux $\hbar|A|^2/mk$ through every sphere (computed symbolically) — conserved outflow, the meaning of "outgoing."
Standing↔travelling dictionary: $j_\ell=\tfrac12(h^{(1)}+h^{(2)})$ — a standing ripple is equal parts arriving and leaving; Wronskian $W[h^{(1)},h^{(2)}]=-2i/x^2$ certifies the pair's independence.
Exterior doctrine: outside any finite interaction region the solution per channel is $\alpha h^{(1)}+\beta h^{(2)}$ ; elastic flux balance forces $|\beta|=|\alpha|$ — the S-matrix's cradle (machinery owned by the scattering chapter, named).
Evanescent continuation: $h_0^{(1)}(i\kappa r)=-e^{-\kappa r}/\kappa r$ exactly — bound-state exteriors are imaginary-argument outgoing waves; deuteron: $\kappa=0.2316\,\text{fm}^{-1}$ , tail $4.32$ fm (computed from $E_B=2.2246$ MeV).
The $\ell=0$ evanescent profile $e^{-\kappa r}/r$ is the Yukawa function — the range-mass dictionary of the portrait.
Key idea: one complex combination turns the standing radial alphabet into travelling waves — and analytic continuation in $k$ turns the same functions into every bound state's tail.

The core insight

$h_\ell^{(1,2)}=j_\ell\pm in_\ell$ are the outgoing/incoming radial waves: asymptotically pure $e^{\pm i(kr-\ell\pi/2)}/kr$ with conserved radial flux, the mandatory exterior basis outside any finite potential — and at imaginary wavenumber $h_\ell^{(1)}(i\kappa r)$ becomes the decaying exponential tail of every bound state.

Where this lives today

Every scattering code that matches at a channel radius — nuclear reaction R-matrix solvers, atomic close-coupling packages, electron-molecule codes — stores its exteriors as Hankel functions, real- $k$ for open channels and imaginary- $k$ for closed ones, exactly the two faces derived here. Antenna and nanophotonics engineering runs the identical mathematics under its opposite sign convention (the warning box's minefield is a daily occupational hazard at that interface): multipole radiation fields are $h_\ell^{(1)}$ 's, and "spherical wave expansions" of measured antenna patterns are Hankel coefficient fits. And in nuclear structure, the halo systematics that began with the deuteron's 4.3 fm tail now organize the drip-line laboratory program: one-neutron halos in exotic isotopes are identified precisely by their anomalously long imaginary-argument Hankel exteriors.

The physical picture

Take the s-wave standing ripple $j_0=\sin(kr)/kr$ and pull it apart the way you learned to pull apart $\sin$ : $\sin kr=\tfrac{1}{2i}(e^{ikr}-e^{-ikr})$ . Half the amplitude is a wave crawling outward, half a wave crawling inward, in permanent balance — that is what "standing" means. The Hankel functions simply name the halves. Nothing new is being solved; the solution space of the radial equation is still two-dimensional per channel, and $\{h^{(1)},h^{(2)}\}$ is a second basis for it, related to $\{j,n\}$ exactly as $\{e^{ix},e^{-ix}\}$ is to $\{\sin,\cos\}$ .

Figure

Same two-dimensional solution space, two bases: standing pair vs travelling pair

Why the travelling basis earns its own article: boundary conditions in open problems are flux statements. "Nothing comes in from infinity except the beam I sent" — "the decay products only leave" — "the detector sits at large $r$ and counts arrivals": every one of these sentences picks out a specific mix of $h^{(1)}$ and $h^{(2)}$ , and none of them can even be phrased in the standing basis without awkward complex recombination. The nineteenth century learned this in acoustics and optics (choosing the radiating solution is exactly Sommerfeld's radiation condition — one honesty line, the electromagnetic version belongs to that track); quantum mechanics inherited the grammar intact.

Construction and credentials

Define, for each $\ell$ :

h_\ell^{(1)}(x)=j_\ell(x)+i\,n_\ell(x),\qquad h_\ell^{(2)}(x)=j_\ell(x)-i\,n_\ell(x) \qquad (1)

where: both inherit "solution of the spherical Bessel equation" from their parents by linearity (verified directly for $\ell\le2$ ), and they are exchanged by complex conjugation — $h^{(2)}=(h^{(1)})^*$ for real $x$ — which is time reversal's fingerprint: conjugating a stationary state reverses its currents, turning leaving into arriving. Every closed form follows from the previous article's (consumed, not re-derived):

h_0^{(1)}=-\frac{ie^{ix}}{x},\qquad h_1^{(1)}=-\Big(\frac1x+\frac{i}{x^2}\Big)e^{ix},\qquad h_2^{(1)}=\Big(\frac{i}{x}-\frac{3}{x^2}-\frac{3i}{x^3}\Big)e^{ix} \qquad (2)

where: the striking structural fact — each is a single $e^{ix}$ times a polynomial in $1/x$ (all three verified symbolically) — is the travelling-wave property made algebraic: no $e^{-ix}$ admixture anywhere, at any radius. The standing pair mixes both exponentials at every $x$ ; the Hankel pair separates them exactly, not just asymptotically.

Derivation AOutgoing means outgoing: asymptotics and the flux certificate

We establish the two credentials that justify the names: the far field is a pure running exponential with the channel phase ladder, and the probability current of an $h^{(1)}$ wave is genuinely, quantitatively outward — conserved through every sphere. The current computation is this article's central three lines of physics.

Step 1. Far field. From the parents' asymptotics (previous article, consumed): $j_\ell\to\sin(x-\ell\pi/2)/x$ and $n_\ell\to-\cos(x-\ell\pi/2)/x$ , so $h_\ell^{(1)}\to[\sin-i\cos](x-\ell\pi/2)/x=-ie^{i(x-\ell\pi/2)}/x=(-i)^{\ell+1}e^{ix}/x$ . Verified sharper than asymptotically: $h_\ell^{(1)}x\,e^{-ix}$ is exactly a polynomial in $1/x$ whose constant term is $(-i)^{\ell+1}$ ( $\ell\le2$ , symbolic). Attach $e^{-iEt/\hbar}$ : surfaces of constant phase obey $kr-\omega t=\text{const}$ , marching outward at speed $\omega/k$ . The mirror statements hold for $h^{(2)}$ with $e^{-ikr}$ .

Step 2. The flux certificate. Take $\psi=A\,h_0^{(1)}(kr)\,Y_0^0$ and compute the radial probability current of Probability Current and the Continuity Equation (its definition, consumed): $j_r=\tfrac{\hbar}{m}\,\text{Im}(\psi^*\partial_r\psi)$ . With $h_0^{(1)}=-ie^{ikr}/kr$ , the modulus is $1/kr$ — so $|\psi|^2$ dilutes as $1/r^2$ — while the phase gradient contributes $k$ ; the product leaves $j_r=\tfrac{\hbar|A|^2}{m}\cdot\tfrac{1}{k r^2}\cdot|Y_0^0|^2$ . Total outflow through the sphere of radius $r$ :

F=\oint j_r\,r^2\,d\Omega=\frac{\hbar\,|A|^2}{m\,k} \qquad (3)

— independent of $r$ (computed symbolically): the $1/r^2$ dilution and the $r^2$ of the sphere's area cancel identically. Probability streams outward at a constant rate through every shell, which is exactly what a source-free, steadily radiating wave must do (the continuity equation's stationary form $\nabla\cdot\mathbf{j}=0$ , honored automatically). Why the dilution is geometry, not absorption: nothing is lost between shells — the same current crosses each one. The $1/r$ amplitude law of every spherical wave (this article's concept-card lineage) is conservation of flux written as an amplitude statement. A detector at 10 cm and one at 10 m see the same total count rate from an isotropic source.

Step 3. Independence bookkeeping: the travelling pair's Wronskian is $W[h^{(1)},h^{(2)}]=-2i/x^2$ (verified $\ell\le2$ ) — nonvanishing everywhere, so $\{h^{(1)},h^{(2)}\}$ spans the solution space exactly as $\{j,n\}$ did, and the dictionary $j_\ell=\tfrac12(h^{(1)}_\ell+h^{(2)}_\ell)$ , $n_\ell=\tfrac{1}{2i}(h^{(1)}_\ell-h^{(2)}_\ell)$ inverts cleanly (verified). Any radial physics can be phrased in either basis; flux questions choose this one. Verified: closed forms (2); SBE membership; prefactor $(-i)^{\ell+1}$ ; dictionary; $W=-2i/x^2$ ; outflow (3) $r$ -independent — all symbolic (article verifier).

The exterior doctrine and the evanescent continuation

Here is the doctrine every matching problem ahead will invoke. Let a potential vanish (or become negligible) beyond some radius $R$ . For $r>R$ the wavefunction obeys the free radial equation, so per channel it is some fixed combination

u_\ell(r>R)\;\propto\;\alpha_\ell\,h_\ell^{(2)}(kr)+\beta_\ell\,h_\ell^{(1)}(kr) \qquad (4)

where: $\alpha_\ell$ meters what arrives, $\beta_\ell$ what leaves — and since the origin is not part of the exterior domain, the singular content of the Hankel pair is legal here (the previous articles' eviction, correctly scoped). Conservation for elastic conditions — nothing absorbed, nothing created — forces $|\beta_\ell|=|\alpha_\ell|$ : the outgoing amplitude can differ from the incoming only by a phase. That phase, channel by channel, is the entire elastic effect of whatever occupies $r<R$ ; packaging it as $\beta_\ell/\alpha_\ell=e^{2i\delta_\ell}$ and organizing the phases into an operator is the business of Phase Shifts and The S-Matrix and Unitarity in Scattering, named here and not performed — this article supplies the stage, those articles play on it. (The base article's matching-to-phase-shift and pole derivations are theirs by chapter ownership; relocation noted in the provenance.)

Go deeper · Derivation B · 2 mink to i-kappa: bound-state tails are outgoing waves in imaginary time

Derivation Bk to i-kappa: bound-state tails are outgoing waves in imaginary time

We continue the outgoing wave to imaginary wavenumber and watch it become the universal bound-state tail. This two-line computation is one of quantum mechanics' most productive analytic continuations — it is the reason "bound states" and "outgoing waves" are two faces of one function family, and it hands the next article its exterior matching functions.

Step 1. A bound state at energy $E=-\hbar^2\kappa^2/2m<0$ has, outside the potential, the same free radial equation with $k^2\to-\kappa^2$ , i.e. $k\to i\kappa$ (the sign of the root chosen so the result decays — the tail-selection doctrine of Boundary Conditions and Normalization of Radial Wavefunctions, consumed). Substitute into the s-wave outgoing form:

h_0^{(1)}(i\kappa r)=-\frac{i\,e^{i(i\kappa r)}}{i\kappa r}=-\frac{e^{-\kappa r}}{\kappa r} \qquad (5)

— exactly the decaying exponential over $r$ (verified symbolically), no approximation, no asymptotics. The incoming function $h^{(2)}(i\kappa r)$ continues instead to $e^{+\kappa r}$ : growth, forbidden. So the boundary condition "decay at infinity" is the boundary condition "outgoing at complex wavenumber" — one condition, two energy sectors.

Step 2. The pattern holds for every $\ell$ : $h_\ell^{(1)}(i\kappa r)$ is $e^{-\kappa r}$ times a polynomial in $1/\kappa r$ (verified for $\ell=1$ : pure decay, no growing piece) — the general exterior wavefunction of an ℓ-wave bound state. These are the functions the next article will match interior solutions onto; handing them over is this derivation's chapter duty.

Step 3. Put a number on a famous tail. The deuteron binds at $E_B=2.2246$ MeV with reduced mass $\mu c^2=469.46$ MeV (computed from $m_p,m_n$ ); its exterior wavenumber is $\kappa=\sqrt{2\mu E_B}/\hbar c=0.2316\,\text{fm}^{-1}$ , tail length $1/\kappa=4.32$ fm (computed) — three times the force's own range. The deuteron mostly lives outside its potential, in the pure $h^{(1)}(i\kappa r)$ region; how much outside is the next article's matching computation (its preview, cashed there). Why physicists carry this continuation everywhere: it unifies the taxonomy. Scattering states, bound states, and (one line of honesty) decaying resonances are the same exterior function at real, imaginary, and complex $k$ respectively — the analytic-in- $k$ picture whose full development, poles and all, is the scattering chapter's Bound States and Zero-Energy Resonances (named; the base's pole derivation is relocated there). *Verified: identity (5) exact; $h_1^{(1)}(i\kappa r)e^{\kappa r}$ rational (pure decay); $\mu$ , $\kappa$ , $1/\kappa=4.318$ fm computed from CODATA masses (article verifier).*

And one observation that earns its portrait. The $\ell=0$ evanescent profile — amplitude $\propto e^{-\kappa r}/r$ — is precisely the Yukawa function. Solving $(\nabla^2-\kappa^2)\phi=-4\pi g\,\delta^3(\mathbf{r})$ , the static field of a point source with a massive mediator, gives $\phi=g\,e^{-\kappa r}/r$ : the same mathematics as (5) because it is the same equation (the free-particle article's smeared-δ audit, rerun at imaginary $k$ — its source-carrying solution now earns its keep). Range and mass trade places through $\kappa=mc/\hbar$ : a force of reach $1/\kappa$ implies a mediator of mass $\hbar\kappa/c$ . Yukawa read the dictionary right to left in 1934 — the portrait tells that story; the pion Compton wavelength $\hbar/m_\pi c=1.41$ fm (computed) against the nuclear force's observed reach is its vindication.

Figure

One function, two energy sectors: h⁽¹⁾ oscillates at real k, decays at imaginary k

Limiting cases

case	result	meaning
$x\to\infty$	$h^{(1)}\to(-i)^{\ell+1}e^{ix}/x$	pure running wave; quarter-turn ladder intact
$x\to0$	$h^{(1)}\approx in_\ell\to-i(2\ell-1)!!/x^{\ell+1}$	singular — exterior use only; interiors keep $j_\ell$
$k\to i\kappa$	$h_0^{(1)}\to-e^{-\kappa r}/\kappa r$	bound-state tail; Yukawa profile at $\ell=0$
$k\to-i\kappa$ (wrong sheet)	$e^{+\kappa r}$ growth	the rejected branch — tail selection is a sheet choice
real superposition $h^{(1)}+h^{(2)}$	$2j_\ell$ : standing wave	travelling pair reassembles the ripple; no flux
elastic exterior	$\|\beta_\ell\|=\|\alpha_\ell\|$ : phase-only response	the S-matrix's cradle (scattering chapter's machinery)

Common trapConvention minefield: which Hankel is outgoing depends on the time convention

The statement " $h^{(1)}$ is outgoing" is married to the physics convention $e^{-i\omega t}$ (equivalently $e^{-iEt/\hbar}$ ). Engineering and much of the optics literature use $e^{+j\omega t}$ — under which $h^{(2)}$ radiates and every $i$ in this article flips sign. Copying a formula between a quantum text and an antennas text without checking the time factor produces waves that converge on their sources, a mistake that survives dimensional analysis and only dies when you compute a current. Second trap, superscripts: some older tables write $h^{(+)},h^{(-)}$ , and a few define them with an extra $\pm i$ prefactor — match against the closed forms (2), not the symbol. Third: at small $x$ the Hankel functions are dominated by their singular parent — never use them to represent a field in a region containing the origin; that is $j_\ell$ 's jurisdiction, and mixing the jurisdictions is the most common error in first matching computations.

Worked examples

Example 1 — certify the outgoing s-wave, end to end. Take $\psi=A\,h_0^{(1)}(kr)Y_0^0e^{-iEt/\hbar}$ and audit its physics.

Step 1. Phase motion: the phase is $kr-\omega t$ (plus constants); constant-phase shells move with $dr/dt=+\omega/k>0$ — outward. The incoming twin flips the sign.

Step 2. Current: $j_r=\tfrac{\hbar}{m}\text{Im}(\psi^*\partial_r\psi)$ evaluates to $\tfrac{\hbar|A|^2}{4\pi mk r^2}$ ; integrating over a sphere gives $F=\hbar|A|^2/mk$ , the same through every shell (computed symbolically) — steady radiation, nothing lost in transit.

Step 3. Bookkeeping check: try the same computation on $j_0=\sin(kr)/kr$ . The function is real, so $\text{Im}(\psi^*\partial_r\psi)=0$ : exactly zero net flux — a standing wave defined by its refusal to transport probability. Answer: outgoing = constant positive $F=\hbar|A|^2/mk$ ; standing = $F\equiv0$ ; the current, not the name, is the certificate.

Example 2 — the deuteron's address. How far outside its own force does the deuteron live?

Step 1. Inputs (computed, not quoted): $\mu c^2=m_pm_n/(m_p+m_n)=469.46$ MeV; $E_B=2.2246$ MeV; $\kappa=\sqrt{2\mu E_B}/\hbar c=0.2316\,\text{fm}^{-1}$ .

Step 2. The exterior wavefunction is $u\propto e^{-\kappa r}$ — the $\ell=0$ imaginary-argument Hankel tail of equation (5) — with decay length $1/\kappa=4.32$ fm (computed). The nuclear force's reach is the pion Compton wavelength $\hbar/m_\pi c=1.41$ fm (computed): the tail is three times longer than the arm that holds it.

Step 3. Read the physics: a state bound by only 2.2 MeV — against a potential tens of MeV deep — hovers barely below the continuum, so its exterior wavenumber is small and its tail enormous. Probability leaks far beyond the force region (the exact exterior fraction requires the interior match — next article's computation, previewed here by its own bank). This "halo" structure, extreme in the deuteron, defines an entire modern field of rare isotopes. Answer: $\kappa=0.2316\,\text{fm}^{-1}$ , tail $4.32$ fm $\approx3\times$ the force range — the deuteron is mostly outside.

Origins · 1 minHideki Yukawa and how this was discovered

Hideki Yukawa

1907–1981 · Kyoto and Osaka

Yukawa's Nobel insight is this article's equation (5), read as a wanted poster. In 1934, a 27-year-old lecturer in Osaka — self-doubting, stuck for years on the problem of what holds nuclei together, and working in a Japan where theoretical physics was barely a generation old — asked the dictionary question: the nuclear force dies off beyond a femtometer or two; what kind of field falls off like $e^{-\kappa r}/r$ ? Massless fields (Coulomb) reach forever; a field whose quantum carries mass satisfies exactly the massive Helmholtz equation whose point-source solution is the imaginary-argument outgoing wave above, with $1/\kappa=\hbar/mc$ the quantum's Compton wavelength. Inverting the observed range gave the mass: about $200$ electron masses, sitting brazenly in the empty gap between electron and proton where no particle was known. He published in a Japanese journal, in English, to near-total silence. Then a 200- $m_e$ particle appeared in cosmic rays (1937 — the muon, famously the wrong one: it ignored nuclei), and the true pion arrived in 1947 photographic emulsions, decaying into that impostor. Yukawa received Japan's first Nobel Prize in 1949, four years after his country's devastation, and spent his later career running Kyoto's new institute and campaigning against nuclear weapons. The full meson theory belongs to the nuclear lanes; what this article owns is his actual calculation's core — a tail length, read as a mass.

History

1868

— Hankel's studies of the cylinder-function pair later named for him

1912

— Sommerfeld's radiation condition: which solution may carry energy to infinity

1926

— Quantum mechanics adopts outgoing-wave boundary conditions for scattering and decay

1934

— Yukawa reads the nuclear force's range as a mediator mass via

e^{-\kappa r}/r

1947

— The pion is found at 140 MeV; the range-mass dictionary confirmed

1949

— Yukawa awarded Japan's first Nobel Prize

Going further

Sakurai & Napolitano Ch. 6 — Hankel functions deployed as the scattering exterior basis.
Griffiths & Schroeter §4.1.3 and Ch. 10 — the family at the source register and its scattering payoff.
Abramowitz & Stegun §10.1 — closed forms and conventions (checked against equation 2).
Jackson, Classical Electrodynamics Ch. 9 — the same functions radiating electromagnetically, opposite conventions and all.

Provenance note: Equations are standard textbook results. A verifier should: (1) confirm the spherical Bessel equation and Rayleigh/Rodrigues formulas, and the closed forms $h_0^{(+)}=-ie^{ix}/x$ , $h_1^{(+)}=-(1/x+i/x^2)e^{ix}$ , $h_2^{(+)}=(i/x-3/x^2-3i/x^3)e^{ix}$ against Abramowitz & Stegun Ch. 10; (2) verify the asymptotic forms $j_\ell\to\sin(x-\ell\pi/2)/x$ , $n_\ell\to-\cos(x-\ell\pi/2)/x$ and hence $h_\ell^{(\pm)}\to\mp i e^{\pm i(x-\ell\pi/2)}/x$ ; (3) recompute the probability-current sign for $h_\ell^{(\pm)}$ ; (4) check the phase-shift matching and hard-sphere limit $\tan\delta_\ell=j_\ell(ka)/n_\ell(ka)$ against Griffiths §11 and Sakurai Ch. 6; (5) verify the Wronskian $W[j_\ell,n_\ell]=1/x^2$ . — Items (1)–(3) are executed in this article's verify script (closed forms symbolic; prefactors $(-i)^{\ell+1}$ extracted exactly; outgoing flux computed with sign and $r$ -independence), plus the evanescent continuation (5)-identity, the travelling-pair Wronskian $-2i/x^2$ , and the deuteron/pion numbers. Item (5) as stated (the $j,n$ Wronskian) is executed in Spherical Bessel and Neumann Functions' verifier, which owns that pair. Item (4) — matching, phase shifts, the hard-sphere formula, and the base's second and third derivations (matching→S-matrix; bound states as poles) — belongs BY NAME to Phase Shifts, The S-Matrix and Unitarity in Scattering, and Bound States and Zero-Energy Resonances per the chapter map (SCAT one-liner discipline); those derivations are RELOCATED there, and this article keeps only the exterior-doctrine statement and the evanescent-tail observation. Edition note: the tail identity hands the exterior functions to Particle in a Spherical Well and Infinite Spherical Box, which performs the actual deuteron matching.

Is anything physically new in h⁽¹⁾, h⁽²⁾ or is this pure repackaging?

Pure repackaging of the solution space — and that is its virtue. New physics enters only through which boundary conditions nature imposes, and those are flux statements ("outgoing only") that the travelling basis expresses in one coefficient. The situation mirrors $\sin,\cos$ versus $e^{\pm ix}$ in one dimension: same span, different questions answered cleanly.

Why does the current computation single out h⁽¹⁾ when the density |ψ|² looks identical for h⁽¹⁾ and h⁽²⁾?

Because density is blind to motion. $|h^{(1)}|^2=|h^{(2)}|^2$ exactly (they are conjugates), so no snapshot of $|\psi|^2$ distinguishes arriving from leaving. The current $\text{Im}(\psi^*\partial_r\psi)$ reads the phase gradient — the part conjugation flips. This is the same lesson Probability Current and the Continuity Equation taught in one dimension: stationary densities can hide steady transport, and only $\mathbf{j}$ testifies.

What happens to the flux certificate for ℓ > 0?

The same $r$ -independence, with the same mechanism: asymptotically $|h_\ell^{(1)}|\to1/kr$ regardless of ℓ (the quarter-turn ladder affects phase, not magnitude), so the outward current through large spheres is again constant. At finite $r$ the $\ell>0$ functions carry extra near-field structure — terms falling faster than $1/r$ that transport no net probability, precisely the "reactive near field" an antenna engineer would name. Only the $1/r$ part radiates; everything else churns in place.

Do the h-functions appear in bound-state problems even though bound states are real functions?

Constantly — via the continuation. At $k=i\kappa$ , $h_\ell^{(1)}(i\kappa r)$ is (up to an overall constant you can choose real) a real decaying function; the complexity of the travelling basis lives on the real- $k$ axis only. So the finite spherical well's exterior, the deuteron tail, and every halo wavefunction are "Hankel functions" with no imaginary part in sight. One family, real where it must be.

How literal is the resonance remark — states at complex k?

Literal, with bookkeeping owed. A decaying quasi-bound state is an outgoing-only solution at $k=k_0-i\gamma$ ( $\gamma>0$ ): the time factor then decays as $e^{-\Gamma t/2\hbar}$ while the spatial tail grows exponentially — alarming at first sight, resolved by causality (the state has only been leaking for a finite time). Making this respectable (Gamow vectors, rigged spaces — one honesty line to the formalism chapter) and extracting widths from pole positions is the scattering chapter's resonance article. The seed is exactly this article's analytic continuation, pushed off the axes.

Is the Yukawa connection more than an analogy?

More: identical mathematics with reinterpreted symbols. The static field of a massive mediator obeys $(\nabla^2-\kappa^2)\phi=-4\pi g\delta^3(\mathbf{r})$ ; the bound-state exterior obeys $(\nabla^2-\kappa^2)\psi=0$ away from the source region. Both are the Helmholtz equation continued to imaginary wavenumber; the point-source solution of the first is the $\ell=0$ member of the second's function family. What is analogy — and stays in the nuclear lanes — is modeling the actual nucleon-nucleon interaction by one-pion exchange; the function is exact, the force model approximate.