21.5 Bohr Predictions, Series Limits, and Model Boundaries

A fixed final level n_f generates a series with lines from all n_i > n_f. As n_i increases, level differences compress and wavelengths approach a finite series limit. The convergence is not arbitrary patterning; it emerges directly from the 1/n^2 energy dependence and is one of Bohr's strongest empirical wins.

At the limit n_i -> infinity, the transition corresponds to ionisation threshold from n_f. This connects discrete lines with continuum edge behavior and gives a clean interpretation of ionisation energy as the energy needed to lift the electron from bound state to E = 0.

For hydrogen ground state, ionisation energy is 13.6 eV. From n = 3 it is much smaller, only 1.51 eV. This scaling explains why highly excited states are weakly bound and easy to ionise, and why plasma and stellar-atmosphere models track level populations carefully.

The model's strength is controlled simplification: it captures the dominant energy structure of one-electron systems with minimal assumptions. Its weakness is missing wavefunction-based probability structure, which is needed for transition probabilities and richer atomic phenomena. Historically, Bohr is therefore best treated as a bridge model between classical and quantum frameworks.