Atomic spectral transition
Enter an atomic number and two different positive principal quantum levels.
About the Rydberg equation
The Rydberg equation describes the wavelengths associated with electron transitions in hydrogen and hydrogen-like ions. An electron in such a system can occupy only particular energy levels identified by the positive integer principal quantum number n. When it moves between two different levels, the atom emits or absorbs a photon whose energy equals the difference between those levels. This calculator turns the atomic number and level numbers into a vacuum wavelength, inverse wavelength, and frequency.
The calculation uses 1/λ = RZ²|1/n₁² − 1/n₂²|, where R is the Rydberg constant, 10,973,731.568160 per meter; Z is atomic number; and n₁ and n₂ are the two principal quantum numbers. The absolute value means the displayed wavelength is positive whether the levels are entered in emission or absorption order. Wavelength is obtained by taking the reciprocal and is converted from meters to nanometers. Frequency follows from f = c/λ, using the exact vacuum speed of light, 299,792,458 meters per second.
For neutral hydrogen, Z is 1. A transition between n = 3 and n = 2 gives the familiar red Balmer-alpha wavelength near 656.1 nanometers. A one-electron helium ion has Z = 2, so its inverse wavelengths are four times those for matching hydrogen level pairs. The simple Z-squared scaling is a defining feature of the ideal hydrogen-like model. Both level entries must be positive integers, and equal levels do not create a transition.
The equation is highly accurate for the idealized case of one electron orbiting a much heavier nucleus, but real spectra contain refinements. Reduced nuclear mass shifts the effective Rydberg constant slightly between isotopes. Fine structure, hyperfine structure, Lamb shifts, external electric or magnetic fields, and nuclear motion can split or displace measured lines. Multi-electron atoms also involve electron shielding and interactions that the elementary equation does not capture.
Spectroscopists use wavelength patterns to identify atoms and ions, examine astronomical sources, calibrate instruments, and connect observed light to quantum energy differences. The named hydrogen series arise when one level remains fixed: Lyman ends at n = 1, Balmer at n = 2, and Paschen at n = 3. Use this result as a canonical vacuum prediction for a hydrogen-like species. For laboratory comparison, also consider air versus vacuum wavelength, isotope, experimental precision, and the physical corrections appropriate to the source.
Rydberg equation FAQ
Does the order of n₁ and n₂ matter?
The calculator takes the absolute energy-level difference, so reversing the levels gives the same wavelength magnitude. The physical direction still determines whether the photon is emitted or absorbed.
Which value of Z should I enter?
Enter the nuclear atomic number, such as 1 for hydrogen or 2 for helium. The formula applies directly only when the species has a single bound electron.
Why must the level numbers be integers?
Principal quantum numbers label discrete bound states and are positive integers in this model. A fractional or zero level does not represent an allowed state.
Is the result measured in air or vacuum?
The result is a vacuum wavelength derived from the vacuum Rydberg constant. Measured air wavelengths differ slightly because air has a refractive index above one.
Why can a measured spectral line differ?
Finite nuclear mass, isotope effects, fine structure, fields, and other quantum corrections can shift or split a line. The calculator provides the canonical nonrelativistic hydrogen-like prediction.