Rydberg formula

The Rydberg formula is a strikingly simple equation that predicts the wavelengths of the spectral lines of hydrogen. Built up in the 1880s by Johann Balmer and Johannes Rydberg from the measured lines alone, it expresses each wavelength using two small whole numbers and one universal constant. With nothing but integers it reproduces the entire ladder of hydrogen's visible, ultraviolet, and infrared lines.

What is remarkable is that this formula was found empirically — by spotting a numerical pattern in the data — decades before anyone understood why it should hold. That a wild collection of colours could be tamed by a rule involving only the squares of integers was a tantalising hint that something deeply countable, something quantized, was at work inside the atom.

The deeper meaning arrived with the Bohr model and then full quantum mechanics. The two integers in the formula are the principal quantum numbers of the electron's initial and final energy levels, and the formula is really a statement that a line's energy equals the difference between two quantized levels. The Rydberg formula thus stands as an empirical law that turned out to encode the quantization of the hydrogen atom, later derived from first principles.