Hydrogen spectrum, Bohr radius and Rydberg energy

The Coulomb Hamiltonian holds three constants: ħ, the reduced mass μ, and the coupling e²/4πε₀, an energy-times-length worth 1.4400 eV⋅nm. There is exactly one way to build a length from them, a₀ = ħ²/(μ ⋅ e²/4πε₀), and evaluating it for a clamped nucleus, μc² = mₑc² ≈ 511 000 eV, with ħc = 197.33 eV⋅nm gives 197.33²/(511 000 × 1.4400) = 0.052918 nm. One natural energy follows, E_R = e²/(8πε₀a₀) = 1.4400/(2 × 0.052918) = 13.606 eV. So before any equation is solved you know atoms are ångström-sized and chemistry is electronvolt-priced; the solution only supplies the integers. Keep both faces of E_R at hand: ħ²/(2μa₀²) reads as the kinetic cost of confinement to a₀, and e²/(8πε₀a₀) as half the Coulomb energy at a₀. Their equality at the same a₀ is the virial theorem in disguise — the atom's size is a genuine compromise between confinement and attraction.

For hydrogen, with its reduced mass, Eₙ = −13.598/n² eV: −13.60, −3.40, −1.51, −0.85, −0.54, … The gaps collapse — 10.20 eV between the first pair, then 1.89 eV, then 0.66 eV — the opposite of the harmonic oscillator's equal spacing, because the Coulomb well keeps widening as it shallows. Two structural facts ride along. The energy depends on n alone, not on l or m, so level n holds n² orbital states (2n² with spin) — a degeneracy special to the 1/r potential. And the ladder has no top rung short of zero: however small ε is, infinitely many levels sit between −ε and 0, because the long 1/r tail always has room for one more distant, barely bound state. Contrast a finite square well, which holds a finite count. E = 0 is not a level but an accumulation point — a warning that something different starts there.

A photon carries a difference: hν = E_R(1/n_f² − 1/nᵢ²), or in wavenumbers 1/λ = R Z²(1/n_f² − 1/nᵢ²). Fixing the floor n_f defines a series. Drops to n_f = 1 form the ultraviolet Lyman series, opening at 13.606 × (1 − 1/4) = 10.20 eV — the 121.6 nm Lyman-α. Drops to n_f = 2 form the Balmer series, whose first four lines at 656, 486, 434 and 410 nm are the visible spectrum of a hydrogen discharge tube. As nᵢ → ∞ each series crowds toward its limit — for Balmer, E_R/4 = 3.40 eV, near 364.6 nm — where the lines pile up faster than any grating resolves them. One bookkeeping trap: the tabulated R∞ = 1.09737 × 10⁷ m⁻¹ belongs to a fictitious clamped nucleus. Hydrogen obeys R_H = 1.09678 × 10⁷ m⁻¹, and the difference displaces every line by one part in 1836 — about 0.36 nm at Balmer-α, enormous against spectroscopy that resolves parts per million.

The μ inside a₀ and E_R is mₑM/(mₑ + M), because the two-body reduction lodged the nuclear recoil there. Writing μ = mₑ/(1 + mₑ/M), hydrogen's levels are a factor 1/(1 + 1/1836.15) = 0.999456 of the clamped-nucleus values. A part-in-1836 correction sounds negligible; against instruments resolving parts per million it is a shout. It is also a discovery tool: the deuteron is twice the proton's mass, so deuterium's correction is half hydrogen's, and each of its lines sits slightly blue of its hydrogen partner — 0.18 nm at Balmer-α, the faint satellite by which Urey identified deuterium in 1932. Pushed to extremes the 'correction' takes over: positronium's partner is a positron, μ = mₑ/2, and the whole ladder floats up to half depth with E_R = 6.80 eV; in muonic hydrogen μ ≈ 186 mₑ, the levels are keV-deep, and the orbit shrinks by the same factor, to about 285 fm, multiplying the muon's overlap with the proton by 186³ ≈ 6 × 10⁶ — which is how that atom measures the proton's radius.

For a one-electron ion the coupling becomes Ze²/4πε₀. The coupling enters a₀ once, inversely, and E_R through its square, so a₀ → a₀/Z and E_R → Z²E_R: Eₙ = −13.606 Z²/n² eV. He⁺ therefore binds 54.4 eV and radiates its 2 → 1 line at 40.8 eV, 30.4 nm — the scaling that unmasked the Pickering series of hot stars, He⁺ drops to n_f = 4 masquerading as hydrogen lines with half-integer n. The same rule runs both directions. Inward: a K-shell electron at Z = 30 moves at v/c ≈ Zα ≈ 0.22, so the relativistic corrections, suppressed by (Zα)², stop being fine at high Z. Outward: a Rydberg atom at n = 100 spans n²a₀ ≈ 0.53 μm — virus-sized — and is bound by 13.606/10⁴ eV = 1.36 meV, more than an order of magnitude below room temperature's kT ≈ 25 meV, which is why such atoms survive only cold, dilute, and shielded from stray fields.

The truncation argument had a single premise: a bound state must decay at infinity to be normalisable, and only E < 0 solutions can. For E > 0 the radial solution oscillates for ever, is admissible at every energy, and describes an electron that arrives, deflects, and leaves. So the spectrum is discrete only below zero — and observation says exactly that. In absorption, series lines march toward the limit, and there the spectrum changes character rather than stopping: any photon above 13.6 eV ionises ground-state hydrogen and gives the surplus to the electron as kinetic energy, so a continuous photoionisation edge extends beyond the last resolvable line. Run backwards, a free electron of any energy can be captured, emitting a continuum photon — the Balmer continuum shortward of 364.6 nm that makes the 'Balmer jump' in nebular and stellar spectra. Bound ladder plus continuum form the complete set of states; drop the continuum and even formal completeness sums come out wrong.