Result

Hydrogen energy levels

E_{n} = - \frac{13.6 eV}{n ^{2}} (n = 1, 2, 3, \dots)

Statement

The allowed energy levels of an electron bound to a proton, labelled by the principal quantum number n.

Symbols and units

$E_{n}$	nth allowed energy level	eV
$n$	principal quantum number	—

When it applies

Hydrogen or hydrogen-like (single-electron) ions only — not multi-electron atoms.

Don't use when

Only for hydrogen or hydrogen-like (single-electron) ions — a multi-electron atom's energy levels are shifted by electron-electron repulsion and this simple formula does not apply.

Derivation

Solving for a particle bound in the Coulomb potential — via the Bohr model's quantized angular momentum, or equivalently the full Schrödinger equation — gives energies scaling as −1/n².

E_{n} = - \frac{13.6 eV}{n ^{2}}

Watch out for

Energies are NEGATIVE, representing a bound state; n = 1 (the ground state) is the MOST negative — most tightly bound — not the highest energy. Level spacing shrinks rapidly with increasing n, which is why the hydrogen spectrum's lines crowd together toward an ionization limit.