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Bohr (1913) proposed that electrons orbit the nucleus only at certain quantized radii where the angular momentum is an integer multiple of ℏ: L = nℏ. Combining this quantization condition with the balance between Coulomb attraction and centripetal acceleration gives the allowed energy levels E_n = −13.6 eV / n². Photons are emitted or absorbed when electrons transition between levels, with hf = E_initial − E_final, reproducing the Rydberg formula exactly for hydrogen.
Derive the quantized radii and energies from the two conditions (circular orbit + angular momentum quantization). Verify that the energy differences reproduce Balmer series wavelengths numerically. Note what the model cannot explain (multi-electron atoms, line intensities, fine structure) to motivate quantum mechanics.
Before 1913, the hydrogen emission spectrum was an empirical puzzle. Experiment showed hydrogen emits light only at specific discrete wavelengths — a pattern summarized by the Rydberg formula — but no one knew why. Classical physics predicted catastrophe: an orbiting electron should radiate energy continuously (accelerating charges emit electromagnetic radiation) and spiral into the nucleus in about 10⁻¹¹ seconds. Atoms clearly did not behave this way, so something was fundamentally missing.
Bohr's solution was audacious: he simply postulated that electrons can only occupy circular orbits where the angular momentum equals an integer multiple of ℏ (L = nℏ). This is not derived from anything more fundamental — it is an assumption. He then combined this quantization rule with the ordinary classical condition that Coulomb attraction provides the centripetal force for a circular orbit. Solving these two equations together yields the allowed radii r_n = n²a₀ (where a₀ ≈ 0.053 nm is the Bohr radius) and the allowed energies E_n = −13.6 eV / n². The integer n, which you will later call the principal quantum number, labels each orbit.
The payoff is the emission rule: when an electron drops from energy level E_i to level E_f, the energy difference is carried away by a single photon with frequency f = (E_i − E_f)/h. Plugging in E_n = −13.6/n² reproduces the Rydberg formula exactly for every series of hydrogen lines — the Lyman series (UV, n→1), Balmer series (visible, n→2), and so on. Predicting these wavelengths from first principles was a triumph no classical model had achieved.
Despite this success, the Bohr model is a semi-classical hybrid that happens to give right answers for hydrogen by a kind of lucky coincidence. It fails completely for helium, cannot predict the relative intensities of spectral lines, and misses the fine structure (small splittings) observed in high-resolution spectroscopy. Most fundamentally, electrons do not actually travel in circular paths — quantum mechanics replaced orbits with probability wavefunctions. The angular momentum quantization that Bohr postulated was later understood by de Broglie as the condition for a standing electron wave to fit around the orbit, and by full quantum mechanics as a consequence of the wavefunction's angular structure.
What survives from Bohr into modern physics is the conceptual framework: discrete energy levels, photon emission and absorption as transitions between those levels, and the ground state as the lowest-energy configuration (not a state of rest). These ideas carry directly into quantum mechanics, spectroscopy, lasers, and atomic clocks. The numerical result E_n = −13.6 eV/n² also remains correct in the full quantum treatment of hydrogen.