Hydrogen Atom Spectrum

Why the hydrogen spectrum matters

For three decades — from Balmer's 1885 fit of visible lines to Bohr's 1913 derivation and Schrödinger's 1926 wave equation — the hydrogen spectrum was the central testbed of quantum physics. A pattern of sharp lines at predictable wavelengths, with no classical explanation, had to be the universe handing in homework. Solving it built quantum mechanics; refining it built QED; mapping it across cosmic distances built modern astronomy.

• Founding evidence for quantization. Atoms emit at sharp frequencies, not a continuous blur. The only theory that accommodates this is one with discrete bound states.
• Astronomy and cosmology. Stellar absorption spectra (Hα as the prominent dark line at 656.3 nm in solar spectra), galactic recession measured by Lyman-α and Balmer line redshifts, the cosmic web mapped via 21 cm hyperfine emission of neutral hydrogen.
• Precision metrology and QED. The 1s → 2s transition at 2,466,061,413,187,035 Hz has been measured to 13 figures — among the most precise measurements in physics. Comparison with QED predictions tests vacuum polarization and probes the proton charge radius.
• Plasma diagnostics. Tokamak fusion plasmas, solar flares, lightning — all glow with hydrogen lines whose widths and ratios reveal temperature, density, and ionization state.
• Pedagogy. Hydrogen is the only atom solvable in closed form. Every QM textbook walks through the Schrödinger equation in spherical coordinates and arrives at the same {n, ℓ, m} quantum numbers and energies. It is the canonical first non-trivial application of the new mechanics.

Bohr energies and the Rydberg formula

The non-relativistic Schrödinger equation for one electron in a Coulomb potential V(r) = −ke²/r yields bound-state energies:

E_n = − m_e · k² · e⁴ / (2ℏ² · n²) = −13.6057 eV / n²,    n = 1, 2, 3, ...

Photon emitted in transition n_i → n_f (with n_i > n_f):

E_γ = E_i − E_f = 13.6 eV · (1/n_f² − 1/n_i²)
λ_γ = hc / E_γ → 1/λ = R_H (1/n_f² − 1/n_i²), R_H = 1.097 × 10⁷ m⁻¹

The Bohr radius a₀ = 5.29 × 10⁻¹¹ m sets the spatial scale; orbital radii scale as ⟨r⟩_n ~ n²·a₀. Higher-n "Rydberg states" extend across thousands of nanometers and ionize easily.

Spectral series

Series	n_f	Region	Strongest line	λ (nm)
Lyman	1	UV	Lyman-α (n=2→1)	121.567
Balmer	2	Visible	Hα (n=3→2)	656.279
			Hβ (n=4→2)	486.135
			Hγ (n=5→2)	434.047
			Hδ (n=6→2)	410.174
Paschen	3	Near-IR	Pα (n=4→3)	1875.1
Brackett	4	Mid-IR	Bα (n=5→4)	4051.2
Pfund	5	Far-IR	(n=6→5)	7459.9

Worked example — computing Hα

Transition n_i = 3 → n_f = 2 (Balmer Hα):

E_γ = 13.6 eV · (1/4 − 1/9)
    = 13.6 eV · (5/36)
    = 1.889 eV

λ = hc / E_γ
  = (1240 eV·nm) / 1.889 eV
  = 656.4 nm   ← deep red

The observed Hα wavelength is 656.279 nm. The small discrepancy (~0.02%) comes from the finite proton mass (reduced mass correction m_e → μ = m_e · m_p / (m_e + m_p)), and from fine and hyperfine structure that split the line slightly.

Fine and hyperfine structure

Fine structure (~α² · 13.6 eV ≈ 7 × 10⁻⁴ eV scale) splits each n into sublevels labeled by total angular momentum j = ℓ ± 1/2. Three corrections combine: relativistic kinetic energy, spin-orbit coupling, and the Darwin term. Spectrum: n=2 splits into 2s₁/₂, 2p₁/₂ (degenerate in non-relativistic Schrödinger but split by QED's Lamb shift) and 2p₃/₂.

Hyperfine structure (~10⁻⁶ eV) couples electron spin to proton spin. The ground state 1s₁/₂ splits into F = 0 (anti-parallel) and F = 1 (parallel), separated by 5.874 μeV = 1.4204 GHz. Transition photon wavelength: 21.106 cm — the famous "21 cm line" used to map neutral hydrogen across galaxies and the cosmic web.

Common mistakes

• Forgetting that energies are negative. E_n = −13.6/n² eV — bound states have negative energy relative to the ionization threshold E = 0. The ionization energy is +13.6 eV, the energy you put in.
• Confusing emission and absorption. Same wavelengths, opposite direction: hot hydrogen emits Hα at 656.3 nm; cool hydrogen between you and a hot continuum absorbs at 656.3 nm. The sun's photosphere shows Hα as a dark Fraunhofer absorption line.
• Mixing the Rydberg constant for hydrogen and ∞. R_∞ ≈ 1.097373 × 10⁷ m⁻¹ assumes an infinitely heavy nucleus; R_H ≈ 1.096776 × 10⁷ m⁻¹ uses the reduced mass — 0.054% smaller.
• Treating each n as one state. The hydrogen level E_n is n²-fold degenerate (in non-relativistic theory): ℓ = 0, 1, ..., n−1, each with 2ℓ+1 m values, times 2 for spin. n=2 has 8 states, all at the same Bohr energy.
• Using Bohr formulas for multi-electron atoms. The −13.6/n² form is exact only for one-electron systems (H, He⁺, Li²⁺, ...). For helium and beyond, electron-electron correlation breaks the simple form; numerical Hartree-Fock or DFT is needed.
• Forgetting selection rules. Not every n_i → n_f is allowed. Electric dipole transitions require Δℓ = ±1 and Δm = 0, ±1. The 2s → 1s transition is forbidden as a single photon dipole (it proceeds via two-photon emission, ~10⁸× slower than 2p → 1s).