Hydrogen spectrum · Bohr's quantized atom

Why does hot hydrogen emit a few sharp colours — a barcode — instead of a smooth rainbow, and what does that say about the atom?

How the lab tests it

From Bohr's postulate that angular momentum is quantized (L = nℏ), derive the energy ladder from first principles — balance the Coulomb pull against the centripetal demand, giving E_n = −μe⁴/(8ε₀²h²n²) = −13.6 eV/n² (μ = the electron–proton reduced mass). A jump n₂→n₁ emits a photon E_{n₂}−E_{n₁}; collect twelve lines from three series (Lyman, Balmer, Paschen) and least-squares fit 1/λ against (1/n₁²−1/n₂²).

What it checks

the Rydberg constant R_H = 1.0968×10⁷ m⁻¹ — recovered as the slope of a SINGLE straight line through the origin (intercept ≈ 0), so all three series obey one law; the Bohr radius a₀ = 5.292×10⁻¹¹ m and ground-state energy E₁ = −13.6 eV, from the constants alone; and the four visible Balmer lines (Hα 656, Hβ 486, Hγ 434, Hδ 410 nm), matching the textbook. Bohr's 1913 model — the first time a quantization rule explained a spectrum, pairing with blackbody and the photoelectric effect as the third pillar of early quantum theory