Bohr Model Energy Levels

Hydrogen's electron cannot hold any energy it likes. It is restricted to a ladder of discrete values indexed by a whole number $n$, and the rung energies are $-13.606\ \mathrm{eV}/n^2$. The sign is doing real work: zero energy is defined as the electron at rest and infinitely far from the proton, so every bound state must lie below that, and the more tightly bound the state, the more negative it is. The $1/n^2$ makes the rungs crowd together as you climb, $-13.606$, $-3.401$, $-1.512$, $-0.850$ eV — converging on zero from below. Freeing the electron from the ground state therefore costs exactly 13.606 eV, and that number is hydrogen's ionisation energy.

Every line in hydrogen's spectrum is a difference between two rungs. The $n = 3 \to 2$ drop releases $13.606\,(1/4 - 1/9) = 1.889$ eV, and dividing 1240 eV·nm by that gives 656.3 nm — the deep red Hα line that gives emission nebulae their colour across half the sky. The $n = 2 \to 1$ drop releases 10.20 eV at 121.6 nm, Lyman alpha, which is ultraviolet and absorbed by the atmosphere. That single fact of arithmetic is why hydrogen's Balmer series has been studied since the 1880s from the ground and the Lyman series needed rockets and satellites.

Niels Bohr published the model in 1913, working from Rutherford's newly discovered nucleus and from Johann Balmer's purely empirical 1885 formula for hydrogen's visible lines. His postulate was that the electron's orbital angular momentum comes only in whole multiples of $\hbar$, which forces particular orbits and hence particular energies. What made it convincing was not the picture but the number: the model produced the Rydberg constant out of $h$, $m_e$, $e$ and $c$, reproducing a quantity that had been measured to several figures and never explained. When quantum mechanics arrived in 1926 it discarded the orbits entirely — no trajectory, no definite radius, only a probability cloud — and yet the energy ladder came through unchanged. That is an unusual outcome in physics: the answer outlived the reasoning that produced it.

The limits are sharp and routinely ignored. This is hydrogen, or more generally a one-electron ion — He⁺, Li²⁺ — where the energies scale as $Z^2$. It does not work for helium or anything else with two or more electrons, because electron–electron repulsion appears nowhere in it; a "Bohr model of carbon" produces numbers with no physical basis whatever. Second, the negative sign is not decorative. The page requires $E_n < 0$ because a positive energy means an unbound electron, which this model does not describe. Third, $n$ is a positive integer, so when you solve backwards for it a result that does not land near a whole number is telling you the input does not correspond to a real level. Fourth, the electron is not orbiting: the ground state has zero orbital angular momentum, which is exactly what Bohr's own postulate forbade. He got the energies right and the mechanism wrong. And finally, keep 13.606 eV distinct from the Rydberg constant $R_\infty = 1.097 \times 10^{7}\ \mathrm{m}^{-1}$ — one is an energy, the other a reciprocal length, they are related by $hcR_\infty$, and they are frequently confused because both are called "the Rydberg".