Discrete Colors
Classical physics said that an atom should be able to emit light of any color. But if you look at hydrogen gas through a prism, you only see four specific lines. This is the Balmer Series.
\[\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{n^2} \right)\]
This formula was a mystery for decades because there was no reason for integers (\(n\)) to appear in the laws of physics.
Worked Examples
Example 1: The Red Line
Using \(n=3\) in the Balmer formula gives the wavelength of the brightest red line in the hydrogen spectrum (\(656 \text{ nm}\)). This precise match between math and nature forced physicists to admit that atoms are "quantized."
The Bridge to Quantum Mechanics
Atomic spectra are the "Music of the Spheres." Just as a guitar string can only play certain notes because of its length, an electron can only have certain energies because of the "size" of the atom. The integers in the Balmer formula are Quantum Numbers. They represent the fact that an electron's wavefunction must "fit" inside the potential well of the nucleus. Everything you see around you—the colors of a sunset, the glow of a LED—is the result of these discrete quantum jumps.