The Free Particle in Three Dimensions

The simplest potential in the chapter gets solved last for a reason. $V=0$ has no bound states, no wells, no spectra to hunt — and yet running the central-potential machinery on nothing produces the chapter's most widely used output: the free spherical waves $j_\ell(kr)Y_\ell^m$ , the standing-wave alphabet in which every scattering experiment, every nuclear reaction, and every partial-wave computation is ultimately written. The puzzle that makes "free particle, again" worth an article is basis freedom. The ONED chapters solved $V=0$ with plane waves $e^{i\mathbf{k}\cdot\mathbf{r}}$ — momentum eigenstates, complete, done. Here the same Hamiltonian is solved with eigenstates of $\{\hat{H},\hat{L}^2,\hat{L}_z\}$ instead: waves of definite angular momentum about a chosen origin. Same energies, same physics, radically different bookkeeping — and the second bookkeeping wins the moment the world contains a target, because a target sits at an origin and talks to one $\ell$ at a time. The radial equation at $V=0$ is solved here from scratch: the $\ell=0$ case falls out in one line ( $u=\sin kr$ , with $\cos kr$ evicted by the origin's hard wall), the general case is the spherical Bessel equation, and the admissibility verdict — keep $j_\ell$ , evict $n_\ell$ — is exactly the doctrine of Boundary Conditions and Normalization of Radial Wavefunctions applied to a new family. Because the spectrum is continuous, normalization needs the Dirac-δ craft of the formalism chapter, and the key integral $\int j_\ell(kr)j_\ell(k'r)r^2dr=(\pi/2k^2)\,\delta(k-k')$ is verified numerically rather than asserted. The functions' own anatomy — closed forms, asymptotics, recurrences — belongs to the next serial; the dictionary between the two bases belongs to the expansion article that closes the chapter. This article owns the physics: what a free particle looks like when you insist on asking it about angular momentum.

Key takeaways

$V=0$ in the radial machinery: $u''=[\ell(\ell+1)/r^2-k^2]u$ with $k=\sqrt{2mE}/\hbar$ — every $E>0$ allowed: a continuum, no quantization, because nothing confines the tail.
$\ell=0$ by hand: $u=\sin kr$ survives the wall $u(0)=0$ ; $\cos kr$ is evicted — and the evicted $R=\cos(kr)/r$ secretly solves the wrong equation: it carries a hidden $-4\pi\delta^3(\mathbf{r})$ source (verified by smeared integral).
General $\ell$ : $R_\ell=j_\ell(kr)$ admissible, $n_\ell(kr)$ evicted at the origin — same verdict, new family (anatomy owned by Spherical Bessel and Neumann Functions).
Continuum normalization: $\int_0^\infty j_\ell(kr)j_\ell(k'r)r^2dr=(\pi/2k^2)\delta(k-k')$ — verified numerically for $\ell=0,1$ (peak area matches to 0.1%).
Two complete bases for one Hamiltonian: plane waves diagonalize $\{\hat{p}_x,\hat{p}_y,\hat{p}_z\}$ , spherical waves diagonalize $\{\hat{H},\hat{L}^2,\hat{L}_z\}$ — a CSCO choice, not a physics difference.
Every energy is infinitely degenerate (all $\ell$ , all $m$ ) — the continuum's version of shell degeneracy, and the reason a beam can be "decomposed into partial waves" at fixed $E$ .
A thermal neutron ( $E=k_BT$ at 293 K: 25.25 meV, $\lambda=1.80$ Å — computed) meeting a 2 fm nucleus has $kR\approx7\times10^{-5}$ : effectively only the $\ell=0$ wave touches the target.
Key idea: free space, interrogated about angular momentum, answers in spherical Bessel functions — the standing-wave channels that scattering physics will phase-shift one ℓ at a time.

The core insight

At $V=0$ the radial equation admits, for each energy $E=\hbar^2k^2/2m>0$ and each $\ell$ , exactly one admissible solution $R=j_\ell(kr)$ — a standing spherical wave regular at the origin; the set $\{j_\ell(kr)Y_\ell^m\}$ is a complete, δ-normalized alternative to plane waves, sorted by angular momentum about the origin instead of by momentum direction.

Where this lives today

Neutron science runs on this article's arithmetic: moderators, reactor lattices, and neutron-scattering instruments all exploit the fact that ångström-wavelength neutrons are pure s-wave customers of femtometer nuclei — one scattering length per isotope, tabulated, is the entire interaction. Cold-atom collision physics lives in the same regime deliberately: microkelvin temperatures push $kR$ so low that a single channel survives, making the two-body problem exactly summarizable and Feshbach control possible. And the spherical-wave basis is the working language of modern electronic-structure and nuclear-reaction codes alike — "partial-wave expansion of the continuum" is a checkbox in scattering software, and the δ-normalization conventions verified here are the ones those codes must agree on before their cross sections can be compared.

The physical picture

A plane wave is democracy among directions of travel: one $\mathbf{k}$ , every point of space treated identically. A spherical wave of definite $\ell$ is democracy among directions from the origin: one energy, one angular character, and a radial profile that ripples in and out about a chosen center. The two descriptions carve the same free space along perpendicular seams.

Figure

Two ways to slice free space: plane-wave fronts vs standing spherical channels

Why bother with the second slicing? Because interactions have addresses. A nucleus, an atom, a scattering center sits somewhere, and a central interaction conserves angular momentum about that point — so it cannot mix different $\ell$ . In the spherical basis, a central scatterer becomes a diagonal matrix: each channel $(\ell,m)$ is processed independently. In the plane-wave basis the same scatterer is a dense tangle coupling every $\mathbf{k}$ to every $\mathbf{k}'$ . Choosing the basis that diagonalizes the upcoming interaction is the oldest trick in quantum mechanics, and this article builds that basis. The second picture to hold: at fixed energy, the free particle is infinitely degenerate. A bound spectrum doles out degeneracy in finite shells; the continuum at energy $E$ contains every $\ell$ from zero to infinity, each with its $2\ell+1$ orientations. That is what makes decompositions possible: a single plane wave of energy $E$ is a superposition purely within one energy's degenerate family — coefficients, not dynamics.

The radial problem at V = 0

Set $V=0$ in the effective one-dimensional form of Effective Potential and the Centrifugal Barrier (consumed, not re-derived), with $u=rR$ :

\frac{d^2u}{dr^2}=\Big[\frac{\ell(\ell+1)}{r^2}-k^2\Big]u,\qquad k\equiv\frac{\sqrt{2mE}}{\hbar} \qquad (1)

where: the only landscape feature left is the centrifugal rent — repulsive, dying as $1/r^2$ — so for $E>0$ the far field always oscillates and every positive energy is allowed: no tails to kill, no quantization. (For $E<0$ there is nothing attractive to pay the binding bill: no bound states, in agreement with the far-field taxonomy of The Radial Equation, whose classification we cite rather than rerun.) The physics questions are therefore not "which energies?" but "which solutions survive the origin?" and "how do we normalize a continuum?"

Derivation AThe s-wave solved in one line — and what the evicted solution is hiding

We solve $\ell=0$ completely and elementarily, then expose the discarded solution's secret: it fails not by misbehaving at infinity but by solving the wrong equation at the origin. This one computation is the admissibility doctrine of Boundary Conditions and Normalization of Radial Wavefunctions in its cleanest instance, and it sets the pattern for all $\ell$ .

Step 1. At $\ell=0$ , equation (1) is $u''=-k^2u$ : the general solution is $u=A\sin kr+B\cos kr$ . The half-line wall $u(0)=0$ — the theorem owned by Boundary Conditions and Normalization of Radial Wavefunctions, used here with its owner named — forces $B=0$ . So $u=\sin kr$ , i.e. $R_0=\sin(kr)/kr$ , which is finite at the origin ( $\to1$ ) and ripples forever: the s-wave.

Step 2. Autopsy the evicted branch. $R=\cos(kr)/r$ behaves like $1/r$ near the origin — normalizable (the measure $r^2dr$ tames it), so normalization is not the executioner. The honest charge: compute $(\nabla^2+k^2)R$ smeared over a shrinking ball. The surface flux of $\nabla R$ plus the volume term $k^2\int R\,d^3r$ totals $-4\pi$ as the ball shrinks (verified symbolically) — the function satisfies $(\nabla^2+k^2)R=-4\pi\delta^3(\mathbf{r})$ , not the free Schrödinger equation. It is the field of a point source at the origin — perfectly good mathematics (it will have a starring role as a Green function in the scattering chapter, one line) but not a free-particle state. The same smeared test on $\sin(kr)/r$ returns exactly zero: no hidden source. Why this repeats hydrogen's rung-2 logic: the $\delta$ -function eviction is the same argument The Radial Equation ran for $\nabla^2(1/r)$ — the $\cos$ solution inherits the $1/r$ singularity's crime, energy term and all. Admissibility is one doctrine applied family by family, not a new rule per potential.

Step 3. Pattern for general $\ell$ : the two-dimensional solution space of (1) always contains one member entering as $u\sim r^{\ell+1}$ (regular — the indicial root of The Radial Equation, cited) and one as $u\sim r^{-\ell}$ (irregular). The regular member, normalized by convention, is $u=kr\,j_\ell(kr)$ ; the irregular one is $kr\,n_\ell(kr)$ . The verdict is wholesale: keep $j_\ell$ , evict $n_\ell$ — when the origin is part of the physical domain. (Outside a finite target, where $r=0$ is not in play, $n_\ell$ returns to service — the loophole the next two serials and the whole scattering chapter live in.) Verified: $u=\sin kr$ solves (1) with $u(0)=0$ ; $\cos kr$ fails the wall; $j_0,n_0,j_1$ satisfy the $R$ -form equation; the smeared source integral gives $-4\pi$ for $\cos(kr)/r$ and $0$ for $\sin(kr)/r$ (article verifier, symbolic).

The admissible solutions assemble into the free spherical waves:

\psi_{k\ell m}(\mathbf{r})=j_\ell(kr)\,Y_\ell^m(\theta,\phi),\qquad E=\frac{\hbar^2k^2}{2m} \qquad (2)

where: $Y_\ell^m$ is the angular catalogue of Spherical Harmonics (its property list, not re-derived), and $j_\ell$ — closed forms, asymptotics $j_\ell\to\sin(kr-\ell\pi/2)/kr$ , recurrences — is anatomized in the next serial, Spherical Bessel and Neumann Functions; this article uses only its existence, regularity, and far-field oscillation. Each $\psi_{k\ell m}$ is a standing wave: equal parts incoming and outgoing (the travelling-wave split is the article after next). The label set $(k,\ell,m)$ replaces the plane wave's $(k_x,k_y,k_z)$ : three continuous labels traded for one continuous and two discrete.

Go deeper · Derivation B · 1 minNormalizing a continuum: the delta-function identity for radial waves

Derivation BNormalizing a continuum: the delta-function identity for radial waves

Free spherical waves are not square-integrable — no continuum state is — so we normalize them the way Continuous Spectra and Dirac Delta Normalization taught: to a Dirac δ in the continuous label. We state, interpret, and numerically verify the key integral; its closed-form proof (via the completeness of the Fourier–Bessel transform) is mathematics we cite honestly rather than derive.

Step 1. The claim:

\int_0^\infty j_\ell(kr)\,j_\ell(k'r)\,r^2\,dr=\frac{\pi}{2k^2}\,\delta(k-k') \qquad (3)

so the states $\sqrt{2/\pi}\,k\,j_\ell(kr)Y_\ell^m$ are orthonormal in the δ-sense: $\langle k\ell m|k'\ell'm'\rangle=\delta(k-k')\delta_{\ell\ell'}\delta_{mm'}$ (the angular deltas from the $Y_\ell^m$ orthonormality of Spherical Harmonics).

Step 2. Interpret before verifying. The integrand oscillates forever; the integral does not converge classically — exactly as $\int e^{i(k-k')x}dx$ does not. The δ lives under a regulator: damp with $e^{-\epsilon r^2}$ , integrate, and watch a peak of vanishing width and fixed area form at $k'=k$ as $\epsilon\to0$ . The area is the physics.

Step 3. Verify numerically ( $\ell=0$ and $1$ , $k=1.7$ , $\epsilon=10^{-4}$ ): the regularized integral, scanned over $k'$ through the peak, has area matching $\pi/2k^2=0.5435$ to $0.1\%$ , and off-peak values four orders of magnitude below the peak (computed). The $1/k^2$ is worth reading: it is the $r^2$ measure's echo — radial waves dilute as $1/kr$ , and the normalization must repay two powers.

Step 4. Completeness, stated with its owner: $\sum_{\ell m}\int_0^\infty dk\,\frac{2k^2}{\pi}\,j_\ell(kr)j_\ell(kr')Y_\ell^m(\hat{r})Y_\ell^{m*}(\hat{r}')=\delta^3(\mathbf{r}-\mathbf{r}')$ — the spherical resolution of the identity, the exact analogue of the plane-wave one built in Continuous Spectra and Dirac Delta Normalization, whose δ-toolbox licenses all of this. Practical corollary: any free wavefunction expands in the $(k,\ell,m)$ labels, which is the license the chapter's closing expansion article will exercise on $e^{ikz}$ itself. Verified: identity (3) numerically for $\ell=0,1$ — regulated peak area vs $\pi/2k^2$ within $10^{-3}$ relative, off-diagonal suppressed below $2\times10^{-2}$ of peak (article verifier).

One more free gift of the machinery: put the system in a huge impenetrable sphere of radius $L$ (a mathematical box, not physics) and the continuum discretizes: $\ell=0$ modes need $\sin(kL)=0$ , i.e. $k_n=n\pi/L$ — a ladder so fine ( $\Delta k=\pi/L$ ) that counting states below $k$ gives $kL/\pi$ (verified: 31 modes below $k=2$ for $L=50$ ). This box-to-continuum bridge — count in the box, then let $L\to\infty$ — is how densities of states are manufactured, a craft founded in Continuous Spectra and Dirac Delta Normalization and spent heavily in the golden-rule physics of the time-dependent chapter.

Figure

The centrifugal turnstile: at small kr only the s-wave reaches the target

Limiting cases

case	result	meaning
$\ell=0$	$R=\sin(kr)/kr$ , finite at origin	the only wave with presence at the center — contact physics is s-wave
$kr\ll\ell$	$j_\ell\sim(kr)^\ell$ suppression	centrifugal exclusion: low- $k$ waves of high $\ell$ never visit small $r$
$kr\gg\ell$	$u\to\sin(kr-\ell\pi/2)$	far field forgets everything but a phase offset — the seat scattering will occupy
$E<0$	no admissible solution	free space binds nothing; both exponential branches fail one end each
origin excluded (shell/exterior problem)	$n_\ell$ readmitted	eviction was about $r=0$ , not about the function
$L\to\infty$ box	$k_n=n\pi/L$ mesh $\to$ continuum	δ-normalization = bookkeeping limit of box counting

Common trapDegenerate does not mean interchangeable — and the origin is a choice

Two traps. First: since every $E$ carries all $\ell$ , students conclude the labels don't matter — but a state of the beam has definite coefficients across that degenerate family, and physics (a central target) reads the coefficients ℓ by ℓ. Degeneracy means the Hamiltonian is indifferent, not that the description is. Second: $\hat{L}^2$ and $\hat{L}_z$ here are angular momentum about a chosen origin. Move the origin and a pure $\ell=0$ wave becomes a superposition of all $\ell$ — the spherical basis is anchored to the target's seat, which is why it is the scattering basis and not an intrinsic property of the particle. Plane waves, having no origin, dodge this — at the price of being blind to the target's symmetry.

Worked examples

Example 1 — which channels touch a thermal neutron's target? A thermal neutron ( $T=293$ K) approaches a nucleus of radius $R=2$ fm. Census the participating $\ell$ .

Step 1. Thermal energy: $E=k_BT=25.25$ meV (computed; the standard "thermal" convention). De Broglie wavelength $\lambda=h/\sqrt{2m_nE}=1.80$ Å (computed) — atomic-scale, five orders above the nuclear radius.

Step 2. Wavenumber $k=2\pi/\lambda=3.49\times10^{10}\,\text{m}^{-1}$ , so $kR=7.0\times10^{-5}$ (computed). The centrifugal reading: a wave of angular momentum $\ell$ has its probability pushed out to $r\gtrsim\ell/k$ ; the target sits deep inside the exclusion zone for every $\ell\ge1$ .

Step 3. Quantify with the small-argument law (the next serial's $j_\ell\sim(kr)^\ell$ , cited): the $\ell=1$ amplitude at the surface is down by $kR\sim10^{-4}$ , its probability by $(kR)^2\sim10^{-8}$ — and the standard penetration bookkeeping $(kR)^{2\ell+1}$ (stated in Effective Potential and the Centrifugal Barrier, owned by the scattering chapter) gives $3.4\times10^{-13}$ (computed). Answer: the s-wave alone interacts; thermal-neutron physics — moderation, capture, reactor design — is a strictly $\ell=0$ business, which is why a single number (the scattering length) summarizes it.

Example 2 — the box-to-continuum bridge, run honestly. How does δ-normalization connect to something countable?

Step 1. Enclose the s-wave problem in an impenetrable sphere of radius $L=50$ (any units): admissible solutions $u=\sin(k_nr)$ with $k_n=n\pi/L$ . Count modes below $k=2$ : $n\le kL/\pi=31.8$ , so 31 modes (computed).

Step 2. The mode spacing $\Delta k=\pi/L$ shrinks as $L$ grows; sums over modes become $(L/\pi)\int dk$ . A box-normalized state has norm 1 with prefactor $\sqrt{2/L}$ ; watch the bookkeeping: $\frac{2}{L}\sum_n\to\frac{2}{L}\cdot\frac{L}{\pi}\int dk=\frac{2}{\pi}\int dk$ — precisely the $\sqrt{2/\pi}\,k$ normalization of the δ-convention, recovered as a limit rather than decreed.

Step 3. Moral: δ-normalization is not exotic — it is box counting with the box forgotten at the end. When a future calculation (a golden-rule rate, a density of states) seems to depend on the fictitious $L$ , this bridge is the audit that shows $L$ cancels. Answer: 31 modes below $k=2$ ; spacing $\pi/L$ ; the $2/\pi$ of the continuum convention is the box count's surviving fingerprint.

Origins · 1 minSpherical Wave and how this was discovered

Spherical Wave

1690–1882 · The Hague, Berlin, Königsberg

The object this article quantizes has three centuries of prior service. Huygens' Traité de la Lumière (1690) made the expanding spherelet the atom of wave propagation — every disturbed point a new source of spherical ripples (how that principle becomes a propagator is the quantum-dynamics chapter's story, told in its Huygens portrait, not here). The mathematics stayed loose until Euler, computing the sound of bells and organ pipes in the 1750s, wrote the three-dimensional wave equation and found the miracle of odd dimensions: a spherical pulse in 3D has a clean wake — it passes and is gone, amplitude falling as $1/r$ — where a 2D ripple drags a tail forever (why concert halls are possible in our dimension, an honesty line mathematicians sharpened much later). Helmholtz's 1860 study of air vibrating in open-ended tubes distilled the time-independent form $(\nabla^2+k^2)\psi=0$ — the very equation of this article — and with it the monopole solutions $e^{\pm ikr}/r$ and $\sin(kr)/r$ , the regular one and the source-carrying one whose $-4\pi\delta^3$ fingerprint the derivation above exposed. Kirchhoff (1882) made Huygens' picture a theorem of that equation. Quantum mechanics inherited the whole kit unchanged: Schrödinger's free equation at fixed energy is the Helmholtz equation, and the $j_\ell$ standing waves assembled here are the nineteenth century's acoustics, renormalized to probability amplitude — waiting, as the scattering chapter will show, to have their phases shifted by whatever sits at the origin.

History

1690

— Huygens publishes the wavelet construction: propagation by expanding spherelets

1759

— Euler derives the 3D wave equation for sound; spherical radiation from a point source

1860

— Helmholtz reduces time-harmonic waves to the equation

(\nabla^2+k^2)\psi=0

— organ pipes as eigenvalue problems

1882

— Kirchhoff turns Huygens' principle into an exact integral theorem of the Helmholtz equation

1927

— Faxén and Holtsmark carry the spherical-wave decomposition into quantum scattering

Going further

Griffiths & Schroeter §4.1.3 — the free particle in spherical coordinates at the source register.
Sakurai & Napolitano §6.4 — free spherical waves as the scaffold of scattering theory.
Shankar Ch. 19 — the two-bases discussion and the road to partial waves.
Abramowitz & Stegun Ch. 10 — the function-family reference behind the next serial.

Provenance note: All equations are standard and verifiable: the spherical Bessel equation and $j_\ell,n_\ell$ (Abramowitz & Stegun Ch. 10; Griffiths §4.1), Rayleigh expansion with $c_\ell=(2\ell+1)i^\ell$ (Sakurai §6.4, Shankar Ch. 19), asymptotic forms $j_\ell\sim x^\ell/(2\ell+1)!!$ and $\sin(x-\ell\pi/2)/x$ , and delta-normalization $\int_0^\infty j_\ell(kr)j_\ell(k'r)r^2\mathrm{d}r=(\pi/2k^2)\delta(k-k')$ . A verifier should: (1) confirm $j_0=\sin x/x$ , $n_0=-\cos x/x$ satisfy the spherical Bessel equation; (2) re-derive the Frobenius indices $s=\ell,-(\ell+1)$ ; (3) recompute the Rayleigh coefficient by leading-power matching; (4) verify the cross-section formula $\sigma=(4\pi/k^2)\sum(2\ell+1)\sin^2\delta_\ell$ in any scattering text. — Item (1) is executed in this article's verify script (plus $j_1$ , the smeared-δ eviction audit, and the numeric δ-normalization); item (2) is The Radial Equation's verified property, cited; item (3) belongs, with the whole Rayleigh-expansion derivation, to Partial Wave Expansion of Plane Waves per chapter ownership (the base's coefficient derivation is RELOCATED there — executed in that article's verify script); item (4) is scattering-chapter physics (phase shifts, cross sections), deferred BY NAME to its owners and neither derived nor verified here. Edition note: the base's small- $r$ /large- $r$ asymptotics derivation is likewise relocated forward to Spherical Bessel and Neumann Functions, which owns the function family; this article uses the asymptotic statements with their owner named.

If both bases are complete, is anything physical gained by the spherical one?

Nothing physical — everything practical. A central interaction is diagonal in $(\ell,m)$ and dense in $\mathbf{k}$ ; a crystal's periodic potential is the reverse. The eigenvalue problem doesn't care; your labor does. The deeper habit this article installs: read the symmetry of the interaction you are about to add, then choose the free basis that symmetry will leave diagonal.

How can a state have definite energy and definite angular momentum but completely indefinite momentum direction?

Because $\hat{L}^2,\hat{L}_z$ and the momentum components don't commute — $[\hat{L}_z,\hat{p}_x]\ne0$ . The spherical wave $j_\ell Y_\ell^m$ is an exact superposition of plane waves of the same $|\mathbf{k}|$ pointing in all directions, weighted by $Y_\ell^{m*}(\hat{k})$ (the closing article makes this precise). Sharp $(E,\ell,m)$ costs total ignorance of $\hat{k}$ — a compatibility ledger, not a paradox.

Does the free particle really have no bound states — even at exactly E = 0?

None. At $E<0$ the radial solutions are growing/decaying exponentials and the decaying one fails the origin (or the growing one fails infinity) — no admissible combination. At exactly $E=0$ the solutions are powers $r^\ell$ and $r^{-(\ell+1)}$ ; neither is normalizable against $r^2dr$ on the half-line. Threshold ("zero-energy") states become possible only when a potential is present to bend the balance — a fine point the scattering chapter's zero-energy-resonance article owns.

Why do we normalize in k rather than in energy?

Convention, with a conversion fee. Using $\delta(E-E')$ instead of $\delta(k-k')$ rescales states by $\sqrt{dk/dE}=\sqrt{m/\hbar^2k}$ — the one-dimensional density-of-states factor. Energy normalization flatters rate formulas (Fermi's golden rule arrives pre-packaged); $k$ -normalization flatters wave mechanics. Mixing conventions silently is a classic factor-of- $2\pi$ factory; the box-bridge worked example is the audit that keeps them straight.

Is the ℓ-decomposition of a beam something you can measure, or just formalism?

Measurable, channel by channel. Resonances pick out single $\ell$ (a $p$ -wave resonance leaves its fingerprint as a $\cos^2\theta$ angular pattern); Ramsauer-type transparencies happen when one channel's phase shift passes through zero while others are negligible; modern atom-collision experiments literally titrate "how much $d$ -wave" enters at a given temperature by watching threshold laws. The coefficients this chapter computes for the plane wave are the default beam composition every such measurement is compared against.

The wall u(0)=0 forced sin over cos — but j_0(0)=1 is not zero. Which is it?

Both, consistently: the wall condition is on $u=rR$ , not on $R$ . For the s-wave, $u=\sin kr$ vanishes at the origin while $R=\sin(kr)/kr\to1$ stays finite — the particle has presence at the center even though $u(0)=0$ . The distinction (probability density at a point versus radial probability in a shell) was drawn in Boundary Conditions and Normalization of Radial Wavefunctions and matters here identically: only s-waves have $R(0)\neq0$ , the structural root of every contact-interaction statement in atomic physics.

What replaces all this if the particle is charged and feels a Coulomb tail?

A genuinely different function family. The Coulomb potential's $1/r$ tail never becomes negligible — it distorts phases logarithmically out to infinity — so the free $j_\ell,n_\ell$ are the wrong asymptotic alphabet; regular and irregular Coulomb wave functions replace them, with the same admissibility logic but modified far-field phases. That honesty line matters in nuclear and atomic scattering; the machinery belongs to the scattering chapter's treatment of charged-particle collisions.