Hydrogen Emission Spectrum: Balmer, Lyman, and Paschen Series

Every element in the universe has a unique "fingerprint" — a set of specific wavelengths of light that it emits when excited. For hydrogen, the simplest atom, these spectral lines follow an elegant mathematical pattern discovered empirically by Johann Balmer in 1885 and generalized by Johannes Rydberg:

$\dfrac{1}{\lambda} = R_{H}\left(\dfrac{1}{n_{f}^{2}} - \dfrac{1}{n_{i}^{2}}\right)$

where $R_{H} = 1.097 \times 10^{7}$ m$^{-1}$ is the Rydberg constant, $n_{i}$ is the initial (higher) energy level, and $n_{f}$ is the final (lower) energy level.

This formula, combined with the Bohr model (Problem 187), reveals that each spectral "series" corresponds to transitions ending at a particular level.

The three main series

Lyman series ($n_{f} = 1$): All transitions down to the ground state. These produce ultraviolet photons because the $n = 1$ level is so deep ($-13.6$ eV) that the energy differences are large.

• $2 \to 1$: $\lambda = 121.6$ nm (UV)
• $3 \to 1$: $\lambda = 102.6$ nm
• $4 \to 1$: $\lambda = 97.3$ nm
• $\infty \to 1$: $\lambda = 91.2$ nm (series limit)

Balmer series ($n_{f} = 2$): Transitions to the first excited state. The first four lines fall in the visible spectrum — these are the lines you see when you look at a hydrogen discharge tube through a spectroscope:

• $3 \to 2$: $\lambda = 656.3$ nm (H-alpha, red)
• $4 \to 2$: $\lambda = 486.1$ nm (H-beta, cyan)
• $5 \to 2$: $\lambda = 434.0$ nm (H-gamma, blue-violet)
• $6 \to 2$: $\lambda = 410.2$ nm (H-delta, violet)
• $\infty \to 2$: $\lambda = 364.6$ nm (series limit, UV)

Paschen series ($n_{f} = 3$): Transitions to $n = 3$. All infrared — invisible to the eye but detectable with IR cameras.

• $4 \to 3$: $\lambda = 1875$ nm
• $5 \to 3$: $\lambda = 1282$ nm
• $\infty \to 3$: $\lambda = 820.4$ nm (series limit)

Why spectroscopy matters

When astronomers point a spectrograph at a star, they see absorption lines — dark lines in the continuous spectrum where specific wavelengths have been absorbed by the star's atmosphere. By matching these lines to laboratory spectra, they can identify which elements are present. The sun's spectrum reveals hydrogen, helium (which was discovered this way — named after Helios, the sun god), iron, calcium, sodium, and dozens more.

The Doppler shift of these lines reveals whether the star is moving toward or away from us, and how fast. The redshift of distant galaxies (all spectral lines shifted to longer wavelengths) is the primary evidence for the expanding universe.

The Rydberg constant and its significance

$R_{H}$ can be derived from fundamental constants:

$R_{H} = \dfrac{m_{e} e^{4}}{8 \varepsilon_{0}^{2} h^{3} c}$

The fact that this derived value matches the experimentally measured value to high precision was one of the great triumphs of the Bohr model and early quantum theory.

Common mistakes

• Confusing emission and absorption. Emission produces bright lines on a dark background (excited gas). Absorption produces dark lines on a bright background (cool gas in front of a hot source). Same wavelengths either way — the physics is time-reversed.
• Forgetting the series limit. As $n_{i} \to \infty$, the lines converge to a limit wavelength: $\lambda_{\text{limit}} = n_{f}^{2}/R_{H}$. Beyond this, the spectrum is continuous (ionization continuum).

Try it in the visualization

Select a series (Lyman, Balmer, or Paschen) to highlight its transitions on the energy level diagram. The spectrum bar below shows where each line falls — Balmer lines appear as colored vertical marks in the visible range; Lyman and Paschen are in the UV and IR (shown as labeled markers). Click individual transitions to see the wavelength and energy computed.