Atomic spectra

Heated solids, liquids and gases generally emit a continuous spectrum of electromagnetic radiation - all wavelengths represented, as seen when white light is passed through a spectrometer. But when an individual gaseous element is excited (by heating or an electrical discharge) and its light examined through a spectrometer, the result is completely different: instead of a continuous spread, only a discrete set of sharply defined wavelengths appears, each element's set being unique to that element, like a fingerprint. This uniqueness is what lets scientists identify the chemical composition of stars, the Sun, and unknown compounds purely from their light.

Emission and absorption. When a gas is heated (or illuminated), electrons absorb energy and jump to a higher stationary state; because excited states have a very short lifetime (about $10^{-8}$ s), the electron quickly falls back to the ground state, re-emitting radiation at the same wavelengths it absorbed - this is emission spectroscopy. Conversely, when white (continuous-spectrum) light is passed through a gas, the wavelengths corresponding to allowed transitions are absorbed, appearing as dark lines against the otherwise continuous bright background - this is the absorption spectrum.

Spectral series formula. For hydrogen, the wavelengths of every spectral line are given by the Rydberg-type formula

where $v$ is the wave number (inverse of wavelength), $R=1.09737\times10^7\ \text{m}^{-1}$ is the Rydberg constant, and $m>n$ are positive integers. Fixing $n$ and letting $m$ range over all larger integers generates one entire named series:

Within each series, successive lines crowd closer together as $m$ increases, approaching a limiting wavelength (the series limit) as $m\to\infty$. …