Rydberg Equation Calculator
Enter the lower and upper energy levels of a hydrogen electron transition and the Rydberg equation calculator returns the wavelength of the emitted spectral line in nanometres.
Wavelength from quantum numbers
Enter the two principal quantum numbers n1 and n2 and the calculator returns the wavelength of the hydrogen spectral line (1/λ = R·(1/n1² − 1/n2²)) in nanometres.
Use whole numbers, n2 > n1
Energy levels are positive integers, and the upper level n2 must be larger than the lower level n1 — otherwise no line is emitted.
What is the Rydberg equation?
The wavelengths of hydrogen's spectral lines
The Rydberg equation calculator predicts the exact wavelengths of light that a hydrogen atom emits or absorbs. When an electron drops from a higher energy level (n2) to a lower one (n1), it releases a photon whose wavelength is set entirely by those two whole numbers and a single physical constant — the Rydberg constant. The formula 1/λ = R·(1/n1² − 1/n2²) turns the pair of principal quantum numbers into a wavelength in nanometres, the number that explains why hydrogen glows with its characteristic red, blue, and violet lines. It is the foundation of atomic spectroscopy and one of the first great triumphs of quantum theory.
Enter the lower level n1 and the upper level n2 to get the wavelength of the hydrogen spectral line in nanometres instantly.
The reciprocal of the wavelength equals the Rydberg constant multiplied by the difference of one over each level squared.
1 / λ = R × (1 / n1² − 1 / n2²)The Rydberg constant R is 1.0973731568160 × 10⁷ per metre. With n1 and n2 as positive integers and n2 greater than n1, the equation gives 1/λ in inverse metres; take the reciprocal for the wavelength in metres, then multiply by 1e9 to read it in nanometres.
Suppose an electron falls from level n2 = 3 to level n1 = 2 — the first line of the Balmer series.
Square each level
n1² = 2² = 4 and n2² = 3² = 9 — the squared quantum numbers.
Take the difference of the reciprocals
1/4 − 1/9 = 0.25 − 0.1111 = 0.1389 — the bracketed term.
Multiply by R and invert
1/λ = 1.097e7 × 0.1389, so λ ≈ 656.11 nm — the red Balmer Hα line.
The wavelength tells you exactly which colour of light the hydrogen atom emits for that transition, and the lower level n1 sorts the lines into named families. Transitions ending at n1 = 1 form the Lyman series in the ultraviolet (the Lyman-alpha line at about 121.5 nm). Transitions ending at n1 = 2 form the Balmer series in visible light — Hα at 656 nm (red), Hβ at 486 nm (blue-green), and Hγ at 434 nm (violet) — the lines you actually see through a spectroscope. Transitions ending at n1 = 3 form the Paschen series in the infrared. The larger the gap between n1 and n2, the more energy the photon carries and the shorter its wavelength; as n2 grows very large the lines bunch toward the series limit, the shortest wavelength a given family can reach. This pattern is why a glowing hydrogen tube shows sharp, predictable lines rather than a smooth rainbow.
The equation is exact for hydrogen, but a few conditions must hold.
Hydrogen-like atoms and integer levels only
This calculator uses the Rydberg equation for hydrogen (and hydrogen-like single-electron ions in spirit); it does not model multi-electron atoms, where electron–electron interactions shift the lines. Both energy levels must be positive whole numbers, and the upper level n2 must exceed the lower level n1 — equal or reversed levels emit no line. The named families follow the lower level: n1 = 1 gives the Lyman series, n1 = 2 the Balmer series, and n1 = 3 the Paschen series.