The Hydrogen Atom: Coulomb Radial Equation

Hydrogen is now the most precisely measured system in science. The 1S–2S interval is known to fifteen digits — measured against atomic clocks in Garching — and comparing it with theory tests QED and even the proton's size: the "proton radius puzzle" of the 2010s was settled largely inside muonic hydrogen, the electron swapped for a muon $207$ times heavier. The orbit shrinks by the reduced-mass factor of $186$ (the proton is no longer effectively infinite), parking the muon close enough to feel nuclear structure. Antihydrogen, trapped at CERN, shows the same 1S–2S line — matter and antimatter obeying the same radial equation, so far to a dozen digits. The high-$n$ limit became a technology: Rydberg atoms with $n \sim 60$ are a few tenths of a micron across, exaggeratedly polarizable, and interact strongly enough to blockade one another across several microns — the mechanism behind neutral-atom quantum computers now running hundreds of qubits. And the $Z$-scaling runs daily in astrophysics: iron stripped to a hydrogenic ion in accretion disks emits its scaled Lyman lines near $6.9\ \mathrm{keV}$, an X-ray barcode read by telescopes to weigh black holes. Balmer's integers, at every scale nature offers.

Separation of variables has already done the angular work — Separation of Variables in Spherical Coordinates showed that the spherical harmonics $Y_\ell^m$ carry all direction-dependence — and what remains is a single function $R(r)$ of radius. The masterstroke is the substitution $u(r) = rR(r)$, built at Effective Potential and the Centrifugal Barrier, which makes the radial equation look exactly like a 1D Schrödinger problem: a particle on the half-line $r > 0$ in an effective potential — the Coulomb funnel pulling in, plus a centrifugal barrier $\hbar^2\ell(\ell+1)/2mr^2$ pushing out for $\ell \geq 1$. All your 1D instincts transfer: bound states sit in the pocket, wavefunctions oscillate where allowed and decay where forbidden, and more nodes means more energy.

The two ends of the half-line dictate everything. Near $r = 0$ the centrifugal term dominates ($1/r^2$ beats $1/r$), forcing $u \sim r^{\ell+1}$ — the regular root of the indicial analysis that The Radial Equation performed, which also showed why the alternative $r^{-\ell}$ is inadmissible. At large $r$ a bound state must die exponentially, $u \sim e^{-\kappa r}$, the tail-selection law owned by Boundary Conditions and Normalization of Radial Wavefunctions — whose shooting experiment already bracketed $E_1 = -13.6\ \mathrm{eV}$ numerically, a number this article now earns with algebra. The solution strategy writes both behaviors in by hand and solves for the interpolation between them. That interpolating function wants to be a polynomial; if its power series ran on forever it would rebuild a growing exponential and destroy the state. Where the series stops — and it can only stop at special energies — is where the spectrum comes from. One habit to carry forward: count nodes. The state $(n, \ell)$ has $n - \ell - 1$ radial nodes, so at fixed $n$, cranking up $\ell$ trades radial wiggles for angular motion. The $2p$ state has no radial nodes at all; the $2s$ has one; both sit at the same energy — a coincidence of the pure Coulomb potential that we will have to explain. Here is that habit made visible: the radial probability $r^2|R_{n\ell}|^2$ for the first three states. The $1s$ peaks at exactly one Bohr radius $a_0$, the $2s$ pays for its single node with a long second hump, and the $2p$ — maximal $\ell$ for its level — is nodeless and pushed off the origin by the centrifugal wall.

And here is the payoff the algebra is aiming for: the energies the boundary conditions will select stack up as $E_n = -13.6\ \mathrm{eV}/n^2$, crowding toward the ionization edge, and every jump between rungs is one line of the spectral barcode — Lyman in the ultraviolet, Balmer in the visible, Paschen in the infrared, a taxonomy mapped in full at The Bohr Energy Spectrum and Rydberg Constant. Keep the ladder in view; the whole derivation is a hunt for why the rungs sit exactly there.

The physical picture set the stage; now write it down. Take the general radial equation for a central potential, install the Coulomb attraction $-e^2/4\pi\varepsilon_0 r$ as $V(r)$, and keep the substitution $u = rR$ that makes the whole thing read like one-dimensional mechanics on the half-line:

where: $u = rR$ — reduced radial function; $m$ — electron mass (strictly the reduced mass $\mu$, a refinement quantified at The Bohr Energy Spectrum and Rydberg Constant); $\ell$ — orbital quantum number; $E < 0$ for bound states. This is the funnel-plus-wall of the hero figure in symbols: the Coulomb term digs the pocket, the $\ell(\ell+1)$ term walls off the origin, and a bound electron must sit at negative energy inside the pocket. Everything that follows is a hunt for the values of $E$ this equation actually tolerates.

With the spectrum won, reassemble the wavefunctions from the three pieces the derivation peeled apart — the $\rho^{\ell+1}$ behavior forced at the origin, the exponential tail forced at infinity, and the terminated polynomial interpolating between them. Restoring physical units through $\kappa = 1/na_0$ gives the full radial eigenfunctions:

where: $N_{n\ell}$ — normalization fixed by $\int_0^\infty |R_{n\ell}|^2 r^2\,dr = 1$; the decay length is $na_0$ — bigger orbits for higher levels; the polynomial degree $n - \ell - 1$ counts the radial nodes. The explicit table of these functions, the Laguerre conventions, and the closed-form $N_{n\ell}$ are owned by Hydrogenic Wavefunctions and Quantum Numbers.

One line completes the family: replace $e^2 \to Ze^2$ and the identical solution covers $\mathrm{He}^+$, $\mathrm{Li}^{2+}$, and every one-electron ion — energies $\times Z^2$, sizes $\times 1/Z$ — the single substitution worked through, with its stellar detective story, at Hydrogenic Ions and Scaling with Nuclear Charge.

1 · The red line of hydrogen (confidence builder)

Compute the wavelength of the $n = 3 \to 2$ transition — the H$\alpha$ line that colors emission nebulae red — and the Balmer-series limit.

Step 1. Transition energy from (5): $\Delta E = 13.6\left(\dfrac{1}{4} - \dfrac{1}{9}\right) = 13.6 \times \dfrac{5}{36} = 1.89\ \mathrm{eV}$.

Step 2. Wavelength: $\lambda = \dfrac{hc}{\Delta E} = \dfrac{1240\ \mathrm{eV\cdot nm}}{1.89\ \mathrm{eV}} = \boxed{656\ \mathrm{nm}}$ — Balmer's red line, to three figures, from a boundary condition. (At the next digit the reduced mass matters: the vacuum wavelength is $656.47\ \mathrm{nm}$ once $m \to \mu$ — the refinement quantified at The Bohr Energy Spectrum and Rydberg Constant.)

Step 3. The series limit is ionization from $n = 2$: $\Delta E = 3.4\ \mathrm{eV}$, so $\lambda = 1240/3.4 = \boxed{365\ \mathrm{nm}}$. All Balmer lines crowd between $656$ and $365\ \mathrm{nm}$ — the barcode's whole visible span, fixed by two numbers.

2 · One state by hand: the 2s from the recursion (exam level)

Use the recursion (4) to construct the complete $2s$ radial function, locate its node, and contrast it with the $2p$.

Step 1. Set the dials. For $n = 2$: $\rho_0 = 2n = 4$ and $\kappa = 1/2a_0$, so $\rho = r/2a_0$. For the $2s$, $\ell = 0$, and the series must terminate at $N = n - \ell - 1 = 1$: a degree-one polynomial.

Step 2. Run the recursion once. With $j = 0$, $\ell = 0$: $c_1 = \dfrac{2(0+0+1) - 4}{(1)(2)}c_0 = -c_0$. With $j = 1$: the numerator $2(1+0+1) - 4 = 0$ kills $c_2$. So $v(\rho) = c_0(1 - \rho)$ — the series terminated exactly where the bookkeeping promised.

Step 3. Assemble the state: $u = \rho\,e^{-\rho}v \propto \rho(1-\rho)e^{-\rho}$, i.e.

The node sits where $v$ vanishes: $\rho = 1$, i.e. $\boxed{r = 2a_0 = 1.06\ \mathrm{\mathring{A}}}$ — the red dot in the probability figure. The decay length is $2a_0$: the $n = 2$ shell reaches twice as far as the ground state.

Step 4. Now the $2p$ ($\ell = 1$): termination at $N = 0$ means $v = c_0$, a constant — the recursion numerator $2(0 + 1 + 1) - 4 = 0$ vanishes before a second term is ever born. So $R_{21} \propto r\,e^{-r/2a_0}$: nodeless, pushed off the origin, peaking (in shell probability) at $4a_0$. Same energy, opposite anatomy — and only the $1/r$ potential lets them tie.

Texts. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §4.2 · Sakurai & Napolitano, Modern Quantum Mechanics, §3.7 · Shankar, Principles of Quantum Mechanics, Ch. 13 · Cohen-Tannoudji, Diu & Laloë, Vol. 1, Ch. VII · Bethe & Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (the classic reference).

Provenance. Equations and substitutions follow Griffiths §4.2 (the $u=\rho^{\ell+1}e^{-\rho}v$ method and recursion) and Sakurai §3.7. A verifier should recompute the energy prefactor $me^4/(2(4\pi\varepsilon_0)^2\hbar^2)=13.6\,\mathrm{eV}$, the Bohr radius $a_0=0.529\,\text{Å}$, the Rydberg constant $R_\infty=1.097\times10^7\,\mathrm{m^{-1}}$, the degeneracy sum $\sum_{\ell=0}^{n-1}(2\ell+1)=n^2$, and $\langle r\rangle=\tfrac{a_0}{2}[3n^2-\ell(\ell+1)]$, all standard textbook values (Griffiths, Shankar Ch. 13, Cohen-Tannoudji Vol. 1 Ch. VII). Edition note (2026-07-08). This article folds the approved v3 pilot into the track as its narrative spine, with the AllQFT base grafted for depth; every spine number was machine re-verified before shipping (41 checks — all clean, with two precision tightenings: muonic hydrogen's orbit shrinks by the reduced-mass factor $186$, not the bare mass ratio $207$; and $n \sim 60$ Rydberg atoms are tenths of a micron across, with blockade radii of microns). The spine's $Z$-scaling derivation ("one substitution, every hydrogenic ion") and its Pickering-lines worked example are relocated to "Hydrogenic Ions and Scaling with Nuclear Charge" (QUANT.HYDR.ZSCAL.0945), which owns the one-electron family; this article keeps the hydrogen ($Z=1$) solve and defers by name. Derivations are recast to consume the chapter's earlier serials (indicial roots, tail selection, the $(2\ell+1)$ theorem) rather than re-derive them.

Prerequisites. The radial equation · separation of variables in spherical coordinates · effective potential and the centrifugal barrier · boundary conditions and normalization of radial wavefunctions. Leads to. The Bohr energy spectrum and Rydberg constant · hydrogenic wavefunctions and quantum numbers · radial probability distribution and expectation values · the Bohr radius and characteristic atomic scales · accidental degeneracy and SO(4) · hydrogenic ions and scaling with nuclear charge.