Atoms emit and absorb photons at characteristic frequencies because their bound electronic states have discrete energies.

If an atom changes from an initial stationary state of energy $E_i$ to a final state $E_f$, energy conservation requires the photon energy to equal the state-energy difference:

$$|E_i-E_f|=h\nu.$$

For emission, the atom moves to a lower-energy state and a photon carries away the released energy. For absorption, an incident photon can promote the atom to a higher-energy state when its energy matches an allowed transition.

Example: a hydrogen spectral line

For ideal nonrelativistic hydrogen,

$$E_n=-\frac{13.6,\mathrm{eV}}{n^2}.$$

A transition from $n=3$ to $n=2$ releases

$$\Delta E=13.6\left(\frac{1}{2^2}-\frac{1}{3^2}\right)\mathrm{eV}\approx1.89,\mathrm{eV}.$$

Using the photon energy-wavelength relation gives a wavelength near

$$656,\mathrm{nm},$$

the visible hydrogen H-alpha line.

Spectra measure energy differences

A collection of emission or absorption lines provides a fingerprint of allowed state-energy differences, not a direct list of absolute energies. Multiple transitions can share the same initial or final state.

The presence of energetically matching states is necessary but not always sufficient for a strong spectral line. Transition intensity also depends on how strongly the electromagnetic field couples the two quantum states; this leads to selection rules.

Real spectral lines have width and structure

Ideal infinitely sharp transitions are broadened by effects such as finite excited-state lifetimes, motion of emitting or absorbing atoms, collisions and instrumental response. Additional interactions can split nominally degenerate levels into nearby components.

Atomic spectroscopy is therefore an experimental bridge from measured photon frequencies to quantum energy structure: spectral positions reveal state-energy differences, while line strengths and shapes reveal additional information about transitions and the environment.