What this calculator does
The Rydberg equation predicts the wavelengths of light emitted when an electron drops between energy levels in hydrogen. It was found empirically from observed spectral lines decades before anyone could explain why it worked.
That explanation came with the Bohr model and then quantum mechanics, which derived the same formula from first principles. It remains one of the clearest demonstrations that atomic energy levels are discrete rather than continuous.
The formula
One over the wavelength equals the Rydberg constant times the atomic number squared, multiplied by the difference between one over the lower level squared and one over the upper level squared.
| Term | Meaning |
|---|---|
| Rydberg constant (R) | 1.0973731568 × 10⁷ m⁻¹, one of the most precisely measured constants in physics. |
| Balmer series | Transitions down to level 2, which fall in the visible spectrum and were the first to be catalogued. |
| Lyman series | Transitions down to level 1, which lie in the ultraviolet. |
The inputs explained
| Field | What to enter |
|---|---|
| Atomic number (Z, hydrogen = 1) | Atomic number. Hydrogen is 1. The formula applies to hydrogen-like ions with a single electron. |
| Lower energy level (n1) | The lower energy level the electron falls to. |
| Higher energy level (n2) | The higher energy level it starts from. It must be larger than n1. |
When to use it
Identifying a spectral line
Calculating the wavelength for a given transition allows an observed line to be matched to its origin.
Understanding the visible hydrogen spectrum
The Balmer series produces the familiar red, blue-green and violet lines seen in a hydrogen discharge tube.
Checking an astronomical measurement
Hydrogen lines are the reference points for redshift measurements, and their rest wavelengths come from this formula.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What wavelengths does the Balmer series produce?
Electrons falling from successively higher levels to level 2.
| From level | Wavelength (nm) | Photon energy |
|---|---|---|
| n = 3 | 656.11 nm | 1.890 eV |
| n = 4 | 486.01 nm | 2.551 eV |
| n = 5 | 433.94 nm | 2.857 eV |
| n = 6 | 410.07 nm | 3.023 eV |
Questions
Why are atomic spectra made of discrete lines?
Because electrons can only occupy certain energy levels. A transition releases exactly the energy difference between two of them, which corresponds to one specific wavelength rather than a continuous range.
What is hydrogen-alpha?
The 3 to 2 transition at about 656 nm, the brightest visible hydrogen line. It is the standard reference for measuring redshifts, and the reason so many astronomical images are taken through an H-alpha filter.
Does this work for other elements?
Only for hydrogen-like species with a single electron, such as singly ionised helium. Multi-electron atoms have electrons that interact with each other, which the formula cannot account for.
Why does the atomic number appear squared?
Because a larger nuclear charge binds the electron more tightly, scaling all the energy levels by Z squared. Singly ionised helium therefore emits lines at a quarter of hydrogen's wavelengths.
For the energy those photons carry, see the photon energy calculator. For how these lines shift with motion, see the redshift calculator.