BOHR ATOM
Only the rungs exist — and the jumps between them are Balmer’s lines, to the nanometer.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A nucleus, an electron, and a scandal: classical physics says an orbiting charge must radiate, spiral inward, and destroy every atom in the universe in a hundred picoseconds. Bohr’s 1913 wager was that only certain rungs exist — and that atoms glow only when the electron changes rungs.
The ladder
The rungs sit at minus 13.6 times Z squared over n squared electron-volts — the R-infinity constant, tabulated, with the nucleus held infinitely heavy and disclosed as such. Radii run the other way, growing as n squared times the Bohr radius. The engine underneath is quantized angular momentum: L equals n times h-bar, held on this page as an identity.
The jump
A drop from the upper rung to the lower releases exactly the rung difference as one photon, and lambda equals hc over that difference. Feed it n-two from three through six down to two, and out fall Balmer’s four visible lines — drawn below at their spectral colors (stylized palette). Optics and quanta, sharing the same light.
The drop
The electron descends — the path is presentational, the endpoints exact — and the ladder pays out its photon at the honest color. Lyman drops land beyond violet in the ultraviolet box; Paschen drops fall below red into the infrared. Every neon sign is an atom paying rung differences.
Audit
Audited: the Rydberg formula computed two ways agrees at machine epsilon over every level pair; Z-squared scaling is bitwise — helium-plus over four and lithium-two-plus over nine return hydrogen exactly; L equals n h-bar re-derived from independent constants to three parts in a trillion; and the correspondence ratio falls as n times two-n-minus-one over twice n-minus-one squared — fifteen eighths at n three, one point-oh-one-five at n one hundred — the quantum ladder flattening into the classical ramp.