Rydberg formula

Johannes Rydberg (1888) generalized empirical formulas for hydrogen spectrum wavelengths:

1/lambda = R (1/n1^2 - 1/n2^2)

R = 1.0974 * 10^7 m^-1 (Rydberg constant for hydrogen). n1, n2 positive integers with n2 > n1. Emission: electron from n2 to n1 emits photon.

Numerical example: H-alpha (Balmer)

Transition n2=3 -> n1=2 (visible red): 1/lambda = 1.524e6 m^-1, lambda = 656.4 nm (red). Frequency 4.567e14 Hz, photon energy 1.89 eV. Tabulated 656.281 nm, error < 0.02%.

Hydrogen spectral series

Notable Balmer lines

• H-alpha (3->2): 656.281 nm (red). Strongest, in H II nebulae.
• H-beta (4->2): 486.135 nm (cyan).
• H-gamma (5->2): 434.047 nm (violet).
• H-delta (6->2): 410.174 nm (deep violet).
• H-epsilon (7->2): 397.007 nm (near UV).

Visible in solar Fraunhofer lines (C, F), nebulae, hot stars.

Hydrogenic atoms (general Z)

For single-electron ions: R_Z = R * Z^2.

1/lambda = R Z^2 (1/n1^2 - 1/n2^2)

• He+ (Z=2): 4x H energies.
• Li2+ (Z=3): 9x H.
• Fe25+ (Z=26): 676x H, astrophysical plasma.

Isotope shift: R depends on reduced mass.

Bohr model derivation

Bohr (1913) postulates:

1. Quantized orbits L = n hbar.
2. Centripetal-Coulomb balance.
3. E_n = -13.6/n^2 eV.
4. E_photon = E_n2 - E_n1 = hf gives Rydberg.

R = m e^4 / (8 epsilon_0^2 h^3 c). Excellent agreement with experiment.

Ionization energy

From n=1 to infinity: E_ion = R h c = 13.605 eV. For Z: 13.605 Z^2 eV. He+: 54.4 eV. Li2+: 122.4 eV.

Limitations

• Hydrogen and hydrogenic only.
• No fine structure (Sommerfeld 1916, Dirac 1928).
• No hyperfine (21 cm line).
• No Zeeman/Stark splitting.
• Lamb shift requires QED.

Astronomical spectroscopy

• 21 cm line: galactic H atomic.
• Quasar Lyman alpha forest.
• H II nebulae: H-alpha emission.
• Stars: Balmer absorption, OBAFGKM classes.
• Solar Fraunhofer C, F.
• Exoplanet transit spectroscopy.
• Cosmology: recombination, CMB.

History

Angstrom 1853: 4 first Balmer lines. Balmer 1885: empirical visible series. Rydberg 1888: generalized formula. Lyman 1906: UV series. Paschen 1908: IR. Bohr 1913: derived R from fundamentals. Sommerfeld 1916: relativistic. Schrodinger 1926: full QM. Dirac 1928: relativistic, predicts antimatter.

Calculator modes

Five modes: lambda from n1, n2; frequency and photon energy; spectral series (first 5 lines + limit); hydrogenic atom with general Z; identify transition from observed lambda (best fit n1, n2). Output includes spectral region and historical series name.

The Bohr model: deriving the Rydberg constant

Rydberg's formula was a triumph of pattern-spotting, but it was purely empirical: it fit the observed hydrogen wavelengths beautifully without explaining why whole numbers should appear at all. That explanation arrived in 1913 with Niels Bohr. He postulated that an electron can only occupy orbits in which its angular momentum is a whole multiple of Planck's constant divided by two pi, quantizing the atom's energy into a ladder of discrete levels given by minus 13.6 electronvolts divided by n squared. A photon is emitted or absorbed only when the electron jumps between two rungs, and its energy is exactly the difference between them.

From this single quantization rule the Rydberg formula falls out with the constant no longer measured but computed. Setting the photon energy equal to the gap between level n2 and level n1 reproduces one over lambda equals R times the difference of inverse squares, and R itself emerges as a combination of the electron mass, charge, Planck's constant and the speed of light. The calculated value matched the spectroscopic one to within experimental error, a stunning confirmation that turned an empirical curiosity into a cornerstone of quantum theory. Later work by Schrodinger and Dirac refined the picture, but Bohr's simple ladder already explained the integers.

Spectral fingerprints: how atoms reveal themselves

Because each element has its own set of energy levels, it emits and absorbs light only at a unique pattern of wavelengths, a spectral fingerprint as distinctive as a barcode. Heat a gas and it glows in sharp emission lines; shine white light through it and the same wavelengths appear as dark absorption lines. This is the foundation of spectroscopy, and it lets chemists identify substances from their light alone. Hydrogen's Balmer lines, which the Rydberg formula describes so precisely, are among the most recognizable signatures in all of science.

The reach of this idea is cosmic. Astronomers cannot visit a star, yet by splitting its light into a spectrum they read off which elements it contains, how hot it is, and how fast it moves. Helium was discovered in the Sun's spectrum in 1868, years before it was found on Earth, and named for the Greek word for sun. A shift of the whole line pattern toward the red reveals a galaxy receding, the Doppler effect that first showed the universe is expanding. The humble hydrogen transitions calculated here, redshifted across billions of light-years, are the very rulers by which cosmologists measure the cosmos.

Updated July 2026