Spectra and Discrete Lines

Pass white light through a glass prism and you get a continuous rainbow — every wavelength from violet to red is present. Now replace the white-light source with a sealed tube of hydrogen gas excited by an electric discharge. The prism spreads the light, but instead of a rainbow you see only a handful of sharp, isolated colored lines against a dark background. Each line corresponds to one precise wavelength; all other wavelengths are simply missing.

This is the signature of an emission spectrum. The reverse experiment — shining a continuous source through cool hydrogen gas — produces an absorption spectrum: a full rainbow crossed by dark gaps at exactly the same wavelengths. The same set of wavelengths that hydrogen emits, it also absorbs.

The hydrogen series

For hydrogen the visible lines were catalogued by Johann Balmer in 1885. He found that their wavelengths fit the compact empirical formula

$\frac{1}{\lambda} = R_H \!\left(\frac{1}{2^2} - \frac{1}{n^2}\right), \qquad n = 3, 4, 5, \ldots,$

where $R_H = 1.0974 \times 10^7 \,\text{m}^{-1}$ is the Rydberg constant for hydrogen, measured directly from the spectra. Substituting $n = 3$ gives the red H $\alpha$ line at $\lambda \approx 656\,\text{nm}$ ; $n = 4$ gives H $\beta$ at $\lambda \approx 486\,\text{nm}$ (blue-green); $n = 5$ gives H $\gamma$ at $\lambda \approx 434\,\text{nm}$ (violet). As $n \to \infty$ the lines crowd together and converge to the series limit at $\lambda = 1/(R_H/4) \approx 365\,\text{nm}$ , the edge of the ultraviolet.

The same pattern repeats in other spectral regions when the denominator $1/2^2$ is replaced by $1/n_1^2$ for different integers $n_1$ . The Lyman series ( $n_1 = 1$ ) falls entirely in the ultraviolet; the Paschen series ( $n_1 = 3$ ) lies in the infrared. The unified expression

$\frac{1}{\lambda} = R_H \!\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \qquad n_2 \gt n_1,$

is the Rydberg formula, and it describes every spectral line of atomic hydrogen with a single constant $R_H$ .

Why discreteness matters

Classical electromagnetic theory predicts that accelerating charges radiate at all frequencies continuously. A classical electron orbiting a proton should spiral inward while emitting a broadening spread of frequencies — yielding a continuous spectrum and a collapsing atom, not sharp lines and stability. The fact that atoms produce only a discrete set of wavelengths is therefore direct experimental evidence that energy inside an atom is not continuously variable. An atom can only emit or absorb a photon whose energy $E = hc/\lambda$ matches the difference between two allowed internal energy levels; all other photon energies are simply not exchanged.

This is the physical content of the discreteness: each spectral line encodes the energy gap between a pair of quantized levels. For hydrogen the allowed energies are

$E_n = -\frac{13.6\,\text{eV}}{n^2}, \qquad n = 1, 2, 3, \ldots,$

and the energy of a photon emitted when the atom falls from level $n_2$ to level $n_1$ is

$E_\text{photon} = E_{n_2} - E_{n_1} = 13.6\,\text{eV}\!\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right).$

Setting $E_\text{photon} = hc/\lambda$ and using $hc = 1240\,\text{eV}\cdot\text{nm}$ reproduces the Rydberg formula with $R_H = 13.6\,\text{eV}/(hc)$ . The agreement between this quantum-mechanical prediction and the spectroscopic measurements is exact to the precision of the experiments.

Discreteness is not a special property of hydrogen. Every element has its own characteristic line pattern — a spectral fingerprint used in astronomy to identify the composition of distant stars from the absorption lines imprinted on their light as it passes through the stellar atmosphere. The universality of spectral discreteness across all atoms is what demanded a new theory of matter, and quantum mechanics is that theory.