The Symmetry of Nature
If light (a wave) can act like a particle, can a particle (like an electron) act like a wave? Louis de Broglie hypothesized that it could. He said every particle has a wavelength:
\[\lambda = \frac{h}{p}\]
Even you have a wavelength, but because your momentum (\(p\)) is so large, your wavelength is too tiny to ever measure.
Worked Examples
Example 1: The Electron Wavelength
For an electron in a typical atom, the de Broglie wavelength is about the same size as the atom itself (\(\approx 10^{-10}\) m). This is why electrons must be treated as waves inside atoms—they are too "spread out" to be considered dots.
The Bridge to Quantum Mechanics
De Broglie's wavelength is what explains Bohr's "Quantized Orbits." An electron orbit is stable only if its circumference is an integer number of de Broglie wavelengths (\(2\pi r = n\lambda\)). This means the electron is a Standing Wave around the nucleus. If it's not a standing wave, it interferes with itself and disappears. This is the ultimate proof that matter is fundamentally wave-like.