Hydrogen Energy Levels Calculator

The hydrogen energy levels calculator computes the energy levels of hydrogen and hydrogen-like atoms.

Energy of the lower level-54.422772 eVlevel n1 — negative because the electron is BOUND, and zero is a free electron at rest, infinitely far away

Transition wavelength30.3756 nmn2 → n1 releases 40.817079 eV — ultraviolet

Photon energy40.817079 eV6.54e-18 J

Energy of the upper level-13.605693 eVlevel n2 — closer to zero, so less tightly bound

Ionisation energy from the lower level54.422772 eVwhat it costs to free the electron completely — 13.60569 eV from the ground state of hydrogen

Orbit radius at the lower level26.4589 pma₀n²/Z — the radius grows as n SQUARED, so highly excited atoms are enormous

Electron speed at the lower level4.37538e+6 m/sαcZ/n — in hydrogen's ground state that is α, about 1/137 of the speed of light. THAT is what the fine-structure constant is

Frequency9.8695 PHz

Wavenumber329,211.947 cm⁻¹spectroscopists quote this rather than wavelength because it is proportional to energy

Series limit22.7817 nmthe shortest wavelength in this series, from n = ∞ — the lines crowd together as they approach it

Scaling with Zenergies go as Z²this hydrogen-like ion with Z = 2 has 4× hydrogen's energies and 0.5× its radii — which is why heavy ions emit X-rays where hydrogen emits visible light

Where the Bohr model failsanything with two electronsit works only for one electron around a nucleus. It gets the hydrogen energies exactly right and then fails completely for helium, because it has no account of electron–electron repulsion or of spin

Reduced-mass correction0.05446% longerthe nucleus is not infinitely heavy, so both orbit their common centre. Everything above uses R∞, the infinite-mass idealisation — for hydrogen the real lines sit about 0.05% further to the red

The formula

E_n = −13.606 Z²/n² eV; 1/λ = RZ²(1/n₁² − 1/n₂²)

Negative energies mean bound

Every level here is negative because the zero of energy is a free electron at rest infinitely far from the nucleus. An electron in an atom has less energy than that, so its energy is negative, and the ionisation energy is simply how far it has to climb to reach zero. Levels get closer together as n rises because the depth goes as 1/n².

Radii go as n²/Z while energies go as Z²/n². A hydrogen-like ion of a heavy element is therefore both far smaller and far more tightly bound — which is why such ions emit X-rays where hydrogen emits visible light.

The fine-structure constant, physically

The electron's speed in the ground state of hydrogen is αc, about 1/137 of the speed of light. That is what α actually is in the Bohr picture, and it is also why the model works as well as it does: hydrogen is only mildly relativistic, so a non-relativistic model gets the energies right to about a part in 10⁵.

Its limits are immediate

The Bohr model reproduces hydrogen's spectrum exactly and then fails completely for helium. It has no account of electron–electron repulsion, no spin, and no explanation of why some transitions are strong and others forbidden. It survives as a teaching device because its energies happen to be right, not because its picture of orbits is.

One correction is worth knowing: the nucleus is not infinitely heavy, so both particles orbit a common centre and the reduced mass shifts every line by about 0.05% in hydrogen. That shift is far larger than spectroscopic precision, and comparing it between hydrogen and its heavier isotope is how deuterium was discovered.