Calculate the wavelength of a hydrogen spectral line from the electron transition levels using the Rydberg equation, plus the frequency, photon energy, and named spectral series.
Inputs
The final (lower) principal quantum number the electron falls to.
The initial (upper) principal quantum number the electron starts from. Must be greater than n₁.
Wavelength
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Enter the lower and upper transition levels above.
Inputs
The final (lower) principal quantum number the electron falls to.
The initial (upper) principal quantum number the electron starts from. Must be greater than n₁.
Frequency & Photon Energy
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Enter the lower and upper transition levels above.
Inputs
Choosing a series fixes n₁ and lists the first five lines (n₂ = n₁+1 … n₁+5).
Series Lines
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Choose a series above to list its first five spectral lines.
Transition
Wavelength
Region
4 min read3 steps7 terms3 examples6 FAQs1/λ = R (1/n1^2 − 1/n2^2)
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.
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Walk-through
How to Use This Calculator
3 steps▸
1
Enter the lower and upper energy levels
On the Wavelength tab, enter n₁ (the level the electron falls to) and n₂ (the level it starts from). n₂ must be greater than n₁ — the electron always drops to a lower energy level and emits a photon.
2
Read the wavelength, frequency, and energy
The Wavelength tab shows λ in nanometres along with the series name. Switch to Frequency & Energy for the same transition's frequency in THz and its photon energy in eV and joules.
3
Browse a whole spectral series
Switch to Series, pick Lyman, Balmer, Paschen, Brackett, or Pfund, and see the first five lines of that series listed with their wavelengths and spectral region.
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Reference
Formula & Methodology
3 formulas▸
Rydberg equation
1/λ = R (1/n1^2 − 1/n2^2)
λ is the wavelength of the photon emitted (or absorbed) when a hydrogen electron transitions between principal quantum levels n1 (lower/final) and n2 (upper/initial), and R is the Rydberg constant, 1.09737 × 10^7 m⁻¹.
Wave number
1/λ = R (1/n1^2 − 1/n2^2)
The left side, 1/λ, is called the wave number — the number of wave cycles per meter. It is the natural quantity the Rydberg equation solves for directly, before converting to a wavelength in nanometres.
Photon frequency and energy
f = c / λ | E = h f = h c / λ
Once λ is known, the photon's frequency follows from f = c/λ (c = speed of light), and its energy from E = hf (h = Planck's constant), usually reported in electronvolts for atomic transitions.
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Glossary
Key Terms Explained
7 terms▸
Rydberg equation ↗The formula 1/λ = R(1/n1² − 1/n2²) that predicts the wavelength of light emitted or absorbed when a hydrogen electron moves between two principal quantum levels.
Hydrogen spectrum ↗The set of discrete wavelengths of light that a hydrogen atom emits or absorbs, produced by electrons jumping between quantized energy levels rather than a continuous range of colors.
Spectral line ↗A single sharp wavelength in an emission or absorption spectrum, corresponding to one specific electron transition between two energy levels.
Balmer / Lyman / Paschen series ↗Named families of hydrogen spectral lines grouped by their lower level n1: Lyman (n1=1, ultraviolet), Balmer (n1=2, visible), and Paschen (n1=3, infrared) are the three most commonly cited series.
Transition ↗The event of an electron jumping from one principal quantum level to another, emitting a photon when it drops to a lower level or absorbing one when it jumps to a higher level.
Rydberg constant (R) ↗An empirical physical constant, R ≈ 1.09737 × 10^7 m⁻¹, that sets the scale of every hydrogen spectral line predicted by the Rydberg equation.
Photon ↗A discrete packet of light energy. Its energy E = hf is fixed by its frequency, so every spectral line corresponds to photons of one exact energy and wavelength.
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Scenarios
Real-World Examples
3 worked examples▸
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Chemistry student
The famous red hydrogen line (Balmer-alpha)
Lower level (n₁) 2Upper level (n₂) 3
1/λ = R(1/4 − 1/9) ≈ 1.524 × 10⁶ m⁻¹, so λ ≈ 656 nm — the deep-red Hα line, the brightest and most famous line in the Balmer series and the reason emission nebulae glow red in astrophotography.
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Astronomy hobbyist
A Lyman-series UV line
Lower level (n₁) 1Upper level (n₂) 2
1/λ = R(1 − 1/4) ≈ 8.23 × 10⁶ m⁻¹, so λ ≈ 121.5 nm — Lyman-alpha, far in the ultraviolet and invisible to the eye, but the strongest single emission line in the entire sky at radio-telescope wavelengths of the early universe.
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Physics student
Photon energy of the Lyman-alpha transition
Lower level (n₁) 1Upper level (n₂) 2
E = hc/λ ≈ 10.2 eV for the same n=2 → n=1 drop — nearly enough on its own to ionize a second hydrogen atom, which is why Lyman-alpha radiation is astrophysically important and biologically damaging.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
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
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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
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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
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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.
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Questions
Frequently Asked Questions
6 questions▸
What is the Rydberg equation formula?+
1/λ = R(1/n1² − 1/n2²), where λ is the emitted or absorbed wavelength, R is the Rydberg constant (≈1.09737 × 10⁷ m⁻¹), n1 is the lower (final) principal quantum level, and n2 is the upper (initial) level, with n2 > n1.
What is the Balmer series?+
The Balmer series is the set of hydrogen spectral lines with n1 = 2 — the electron falls to the second energy level. It is the only hydrogen series with lines in the visible part of the spectrum, which is why it was discovered first, in 1885, before the general Rydberg formula existed.
What is the wavelength of the Hα (H-alpha) line?+
Hα is the n2=3 → n1=2 transition in the Balmer series, with a wavelength of about 656 nm — a deep red visible light. It's the strongest hydrogen emission line and is widely used in astronomy to image nebulae and stellar activity.
What is the Lyman series and why is it ultraviolet?+
The Lyman series has n1 = 1 — every transition ends at the ground state, the deepest energy level in hydrogen. Because the energy gap to the ground state is large, every Lyman-series photon has high energy and short wavelength, placing the entire series in the ultraviolet, invisible to the human eye.
What units does the calculator use?+
n1 and n2 are unitless positive integers (principal quantum numbers). Wavelength is reported in nanometres (nm), frequency in terahertz (THz), and photon energy in both electronvolts (eV) and joules (J) — eV is the conventional unit for atomic-scale transition energies.
How is photon energy related to wavelength?+
Energy and wavelength are inversely related through E = hc/λ, where h is Planck's constant and c is the speed of light. Shorter-wavelength photons (like the ultraviolet Lyman series) carry more energy per photon than longer-wavelength ones (like the infrared Paschen series).
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