The Rydberg equation was one of the first quantitative laws in atomic physics, correctly predicting hydrogen's spectral lines decades before quantum mechanics explained why they exist. This calculator computes the wavelength, frequency, and photon energy for any electron transition between two hydrogen energy levels, and names the spectral series it falls into.
How the Rydberg Equation Calculator works
A hydrogen electron can only occupy specific, quantized energy levels labeled by the principal quantum number n = 1, 2, 3, .... When an electron falls from a higher level n2 to a lower level n1, it emits a photon whose wavelength is given by 1/λ = R(1/n1² − 1/n2²), where R is the Rydberg constant (≈1.09737 × 10⁷ m⁻¹). The same formula, run in reverse, gives the wavelength a hydrogen atom will absorb to excite an electron from n1 up to n2. Once λ is known, standard wave-photon relations (f = c/λ and E = hf) give the transition's frequency and energy.
Inputs and what they mean
n1 is the lower, final energy level the electron lands on — it must be a positive whole number. n2 is the upper, initial level the electron starts from, and must be strictly greater than n1 (an electron cannot fall from a level to itself or to a higher one and still emit light). On the Series tab, choosing a named series simply fixes n1 (1 for Lyman, 2 for Balmer, 3 for Paschen, 4 for Brackett, 5 for Pfund) and lists the wavelengths for the next five values of n2.
Limits and edge cases
This calculator uses the simple Rydberg formula for hydrogen (a single proton and electron); it does not apply directly to multi-electron atoms, where electron-electron repulsion shifts energy levels away from the clean 1/n² pattern. It also treats the Rydberg constant as fixed rather than correcting for the small reduced-mass difference between hydrogen and heavier hydrogen-like ions (He⁺, Li²⁺), so results are most accurate for ordinary hydrogen. As n1 and n2 grow large, wavelengths converge toward a series limit (n1²/R) rather than continuing to increase without bound — this is expected physics, not a calculator error. For general photon energy/frequency/wavelength conversions outside the hydrogen-spectrum context, see the Photon Energy Calculator.