De Broglie's Explanation

Chapter 14 established that matter, not just light, has a dual wave-particle nature: every material particle has an associated wave, with wavelength given by the de Broglie relation $\lambda=h/p$. De Broglie's key suggestion was to stop picturing the orbiting electrons inside an atom as tiny particles travelling along a track, and instead picture each one as a STANDING WAVE wrapped around its circular orbit -- directly analogous to the standing waves that form on a fixed string or inside a closed pipe (Chapter 6), which can only vibrate at certain special (resonant) wavelengths.

For a standing wave to exist stably around a closed circular path, the wave must join up smoothly and consistently with itself after one full trip around the circle -- meaning the orbit's total circumference must contain a WHOLE number of wavelengths, with no partial wavelength left over to cause destructive interference. For the nth orbit (radius $r_n$), this condition is written $2\pi r_n=n\lambda_n$, where n = 1, 2, 3, ... must be a positive integer. …