Hydrogen Energy Levels Calculator — The Ladder

Hydrogen’s electron lives on a ladder of allowed energies, E = −13.6 ÷ n² electron-volts, and nowhere in between. Enter two levels, and this calculator gives both energies, the gap between them, and the wavelength of the photon emitted or absorbed in the jump.

The Ladder and Its Rungs

Hydrogen's bound states crowd toward freedom: −13.6 eV at the ground floor, −3.40, −1.51, −0.85, … accumulating at 0, the ionisation limit, beyond which the electron is free and energies run continuous. The spacing law E_n = −13.6 ÷ n² means the ladder's rungs halve and halve again in energy — jumps from high levels are cheap, infrared whispers, while the ground-floor jumps (the Lyman series) are the expensive ultraviolet. Every spectral series is a floor of this ladder: Lyman falls to n = 1, Balmer to 2 (the visible set), Paschen to 3, Brackett and Pfund deeper into the infrared. The calculator's two fields let the reader walk any pair of rungs and read off the photon — the whole of hydrogen spectroscopy reduced to the ladder's statement, with the Rydberg page offering the direct wavelength form for those who prefer it.

Worked Example: H-Alpha, the Red Line of the Universe

The defaults set the most observed transition in astronomy: n = 3 → n = 2. E(3) = −1.511 eV, E(2) = −3.400 eV, and the photon carries the difference, 1.889 eV = 656.4 nm — H-alpha, the deep red that colours every emission nebula photograph, the line through which solar prominences are watched and the star-forming galaxies of the redshift page are detected at cosmic distances. The calculator's 656.4 (with the rounded 13.6 constant) stands a tenth of a nanometre from the laboratory 656.3 — the Rydberg page, carrying more digits, lands it exactly; the two pages' near-agreement is this wave's internal audit, performed in the open. Try n = 4 → 2 for H-beta's blue-green (486 nm) and n = 2 → 1 for Lyman-alpha's ultraviolet 121.6 nm, the most abundant line in the universe's own accounting.

From Levels to Everything Else

The ladder is the seed crystal of atomic physics: the Bohr page derives its rungs from quantised angular momentum; the Rydberg page compresses them into a single reciprocal formula; stellar spectroscopy (the Cepheid and classification literatures) reads temperatures from which rungs a star's hydrogen populates (the Balmer lines peak in strength at A-type stars near 10,000 K, where the n = 2 rung is well stocked but not yet ionised away — the ladder as a thermometer). Even this hub's redshift page depends on it: the identifiable lines whose stretching measures the universe's expansion are these rungs' photons, recognised at unfamiliar wavelengths by their ladder-pattern spacing. A calculator with two integer inputs is a modest thing; the ladder it tabulates underwrites half of astronomy. It also underwrites the unity of this hub's quantum wave: the Bohr page argues the rungs from a postulate, the Rydberg page compresses them into a wavelength formula, the photon page prices their jumps as currency, and this page simply lists them — the four views of one staircase, each calculator a window on the same flight of steps, agreeing with each other to the digits they print because the staircase is real.

The 10.2 eV Gap and the Universe's First Light

The ladder's first step — ground to n = 2, 10.2 eV, Lyman-alpha's price — is cosmology's gatekeeper: in the early universe, hydrogen could not settle into its ground state efficiently while Lyman-alpha photons kept re-exciting it, and the cosmic dark ages' end is written in this rung's traffic (the first stars' Lyman-alpha glow is what the newest telescopes strain to see, redshifted per the hub's other page into the infrared). Closer to home, the same 10.2 eV sets fluorescence lamps' ultraviolet budget. Run 2 → 1 in the calculator for the wavelength (121.6 nm) and you hold the photon that structured the young universe's escape from darkness.

The Ladder in the Discharge Tube

The worked example is not an abstraction for anyone who has wired a hydrogen discharge tube: feed it a few kilovolts and it glows the precise salmon-pink of H-alpha dominating the Balmer set — n = 3 → 2 outshining its siblings because the discharge's electron energies stock n = 3 generously. Grating the glow fans it into the series the calculator tabs: red, blue-green, violet, and the crowd toward the series limit at 364.6 nm. School laboratories buy the tube for a few pounds and the ladder stands visibly in the air, which is more empirical immediacy than most fundamental structures ever offer. The calculator numbers what the tube shows.

Related Tools

More from the Physics Calculators hub: the Bohr Model Calculator — the radii and speeds belonging to these levels — and the Energy to Wavelength Calculator, for photons from any source, not just hydrogen's ladder.

Frequently Asked Questions

Why are all the energies negative?

Because zero is defined as the electron at rest at infinity — free — and a bound electron has less energy than that: you must pay 13.6 eV to evict the ground-state electron, so its account stands at −13.6. The convention is bookkeeping with physical meaning: the negative of the level's energy is its binding energy, the price of ionisation from that rung (from n = 2, only 3.40 eV — which is why Balmer-level hydrogen ionises in modest ultraviolet). The calculator prints the signed levels and the positive photon energy, the two conventions side by side, as the literature does.

What happens at n = infinity, exactly?

The ladder's top: E = 0, the electron free with no kinetic energy to spare — the ionisation threshold itself. Photons above 13.6 eV (shorter than 91.2 nm, the Lyman limit of the Rydberg page) ionise ground-state hydrogen outright, their excess becoming the freed electron's kinetic energy; this continuum absorption is why the ultraviolet sky is opaque with hydrogen and why the ionosphere exists. The calculator's integer inputs stop at finite rungs; this FAQ marks the rung at the top of the stairs, where the discrete becomes continuous and the ladder page's jurisdiction ends.

How were the levels known before any theory?

By measuring the lines and noticing the pattern: Balmer's 1885 formula fit the visible wavelengths from a numerologist's recipe (n² ÷ (n² − 4) in essence), Rydberg generalised it across series in 1888, and Ritz's combination principle (1908) observed that every line is a difference of two terms — the ladder's rungs announcing themselves in the data a generation before Bohr explained them. The calculator's level energies are Ritz's terms in eV; the history matters because it shows the ladder was discovered empirically, not imposed theoretically — the theory's job was to explain a staircase the spectroscopists had already climbed.

Do other elements have ladders this simple?

No — hydrogen's single electron is the unique clean case; add a second electron (helium) and the levels split and multiply under the electrons' mutual repulsion and the exclusion principle, and by iron the level structure is a thicket that fills reference volumes. The simplifications that survive: hydrogen-like ions (He⁺, Li²⁺) keep the 1 ÷ n² ladder scaled by Z², and every atom's outermost electron at high n behaves hydrogenically (a Rydberg atom again). This calculator's jurisdiction is the clean case stated on its door; the periodic table's complexity begins exactly where it ends.

What is the ground state, and can the electron ever fall below it?

n = 1, at −13.6 eV, is the floor — there is no lower rung, which is the atom's stability explained at last: the electron cannot radiate its way to the nucleus because no state exists for it to land in below the ground state, and the ground state itself is stationary. Classical physics had no floor and predicted collapse; the ladder's bottom rung is the universe's atoms' foundation. The calculator requires the lower level to be at least 1 in honour of the fact: the input validation is the physics.

What is a forbidden transition, then?

The ladder's rungs permit jumps selectively: electric-dipole selection rules (Δl = ±1 in the full theory) make some pairs fast (nanoseconds) and others "forbidden" — metastable states that linger seconds to aeons before falling, emitting the green 557.7 nm auroral line and the nebular lines that once invented the phantom element "nebulium" until Bowen identified them as forbidden oxygen transitions in 1927. The calculator reports the energy and wavelength of any pair you enter; which pairs nature actually performs promptly is the selection rules' separate volume, flagged here because a wavelength in the output is a permission of energy bookkeeping, not a promise of a bright line.

How do astronomers use n = 2 specifically?

The Balmer rung is astronomy's workhorse: its population (set by temperature through Boltzmann's factor on this page's −3.40 eV) gauges stellar temperatures, its lines' widths gauge surface gravities and pressures, and H-alpha — this page's worked example — traces ionised gas across the cosmos, from Orion's nursery to galaxies whose light left them when the universe was young. That one rung's occupancy and emissions carry so much of the science is a consequence of the ladder's spacing: 10.2 eV above the floor is a price stellar atmospheres around 10,000 K pay readily, and the universe is full of such atmospheres paying it.