21.2 Discrete Energy Levels and Emission Spectra

If atomic energy were continuous, emitted light from excited gas would form a smooth rainbow. Instead we measure discrete bright lines, each at one precise wavelength. That means atoms occupy specific allowed energies and transition only between those allowed states. Line spectra are therefore direct evidence for quantisation.

For hydrogen, Bohr's model gives a compact formula for level energy. The level index n labels states from ground state (n = 1) upward. More weakly bound states lie closer to zero energy, and ionisation corresponds to reaching E = 0 from below.

Because E_n is negative, moving to lower n makes energy more negative and releases energy as radiation. The emitted photon's wavelength is inversely proportional to transition energy. Large energy gaps give short wavelengths (often ultraviolet), while smaller gaps give longer wavelengths (visible or infrared).

Always convert units carefully. Energy-level formulas are often in electron-volts, but Planck's equation in SI uses joules. Converting eV to J is a common exam checkpoint because one arithmetic slip can shift wavelength by large factors.

Fixing the final level defines a spectral series (for example Balmer with n_f = 2). As initial n increases, successive level gaps shrink, so line spacing decreases and lines converge near a series limit. This is another quantisation fingerprint: non-uniform spacing reflects the 1/n^2 energy structure.