Let the electron live only on quantized orbits and one number decides everything: En = −13.6 eV / n². That single rule explains why hydrogen does not glow in a smear but in sharp lines — a jump from orbit ni to nf radiates a photon of energy |Ei−Ef|, and the Balmer 3→2 line lands at 656 nm, the red of every hydrogen lamp. Energies go in, the ladder is climbed, wavelengths come out. Blue builds it; red breaks it.
source N. Bohr, On the Constitution of Atoms and Molecules (Part I), Phil. Mag. S6, 26, 1–25 (1913) — doi:10.1080/14786441308634955. Rendered, not quoted.
One postulate, one formula: bound states sit at En = −13.6 eV / n² — negative because bound, rising toward 0 (ionization) as n→∞. The orbit obeys L = n·ħ and rn = n²·a₀.
Live, from the pure functions (not looked up):
| n | En (eV) | rn/a₀ | L/ħ |
|---|
Bohr asserted the quantization L = n·ħ — it worked, but why was left open. Eleven years on, the-de-broglie-wavelength answers it: an orbit is stable only when its circumference holds a whole number of matter-waves, 2πr = nλ with λ = h/p. That reproduces L = n·ħ exactly.
So this sphere's ad-hoc rule AMBER is the next sphere's standing wave — each is the other's premise, and Schrödinger later makes it a full wave equation.
The blue team's live re-check: recompute E₁, E₂, the −1/n² scaling and the Balmer 3→2 wavelength from the engine and confirm them against the known truth. If red tampers, this badge turns red.
A hydrogen transition is two integers: an upper level ni the electron falls from, and a lower level nf it falls to. The lower level names the series:
| nf | series | lands in |
|---|---|---|
| 1 | Lyman | ultraviolet |
| 2 | Balmer | visible |
| 3 | Paschen | infrared |
Constants fed in: 13.6 eV (Rydberg energy), hc = 1239.84 eV·nm, a₀ = 0.0529 nm, ħ. That is the whole input to the panel below.
A jump ni→nf emits one photon of energy Ei−Ef. The Balmer 3→2 jump is hydrogen's red line.
Levels drawn at −13.6/n²; the arrow is the chosen jump; the bar at right is the photon's true colour. Every number is computed on the spot, never looked up.
What the machine produces, proven: the Rydberg formula 1/λ = R(1/nf² − 1/ni²) falls straight out of the ladder, the whole Balmer series lands in the visible (3→2 = 656 nm, 4→2 = 486 nm, limit 364.7 nm), and the ionization energy from the ground state is exactly 13.6 eV.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
The quantization L = n·ħ is postulated, not derived — de Broglie explains it, but the full picture needs quantum mechanics. Bohr is not the atom; it is the first model that made the spectrum computable.
"Electrons orbit the nucleus like planets." Cut. The orbit is a bookkeeping fiction; the correct object is a probability cloud (Schrödinger). The energies Bohr got right; the paths he did not.
"En = −13.6/n gives the spectrum." Cut. It is −13.6/n². The square is the whole point — window 6 deletes it and the 656 nm line breaks.
"Bohr explains all atoms." Kept, corrected. Exactly one-electron systems (H, He⁺, Li²⁺…). For the rest it is qualitatively wrong.
The red team's move: change the law to En = −13.6/n (1/n, not 1/n²) and try to slip it past. The blue team's witness (window 7) is watching.
Drop the square and E₂ becomes −6.8 (not −3.4), the level spacings collapse, and the Balmer 3→2 line moves off 656 nm — the witness recomputes, disagrees with the known values, and turns red. Nothing is faked; the attack is real and it is caught.