T-26 · The hydrogen atom

One proton, one electron.

The simplest atom has levels −1/n², so every line in its spectrum is a fraction of one constant, and where the levels form a Pythagorean triple the fraction is a perfect square. Its orbitals are polynomials times an exponential, and their averages are fractions too. Bohr's orbits are the waves that fit.

Every line a fraction

The levels are −1/n² of the Rydberg energy, so a fall from n to the series' floor gives off light of wavenumber R(1/n₁² − 1/n²): a fraction, exactly, times one measured constant.

fall

1/n₁² − 1/n²

λ, nm

3 → 2

5/36

…

4 → 2

3/16

…

5 → 2

21/100

…

6 → 2

2/9

…

7 → 2

45/196

…

Balmer's formula

In 1885, before any of this was understood, Balmer fitted the four visible lines with λ = B n²/(n² − 4). It is the same fraction turned over: B = 4/R.

B = 4/R	…
Balmer against Rydberg	residual …

Pythagoras in the spectrum

When n₁, m and n form a Pythagorean triple, the line's fraction is a perfect square: 1/n₁² − 1/n² = (n² − n₁²)/(n₁n)² = (m/n₁n)². No triple has 1 or 2 as a leg, so the Lyman and Balmer series have none.

squares here	none: 2 is not a leg of any triple