Quantisation of angular momentum and de Broglie concept

De Broglie's concept does more than just assign a wavelength to the electron - it gives a genuine physical justification for Bohr's angular-momentum quantisation rule, something Bohr himself could never supply.

Deriving Bohr's rule from a standing wave. Picture the electron's wave wrapped around its circular orbit. For this wave to exist stably - to reinforce itself on every trip around rather than cancelling itself out - the wave has to join up smoothly with itself after one full circuit; that is, the orbit's circumference must equal a whole number of electron wavelengths, never a fractional number. If the circumference were NOT an integer number of wavelengths, successive loops of the wave would fall increasingly out of step and destructively interfere, so no stable wave (and therefore no stable orbit) could exist there at all.

Writing this condition down,

and substituting de Broglie's own relation $\lambda=h/mv$ from equation (2.9),

Rearranging,

i.e. angular momentum $=nh/2\pi$. This is exactly Bohr's third postulate, equation (2.1), reproduced here from a completely independent starting point - the requirement of a stable standing wave. That the two approaches agree perfectly is powerful evidence that de Broglie and Bohr were describing the same underlying physical truth from two different angles, and it is what finally supplied the missing "why" behind Bohr's quantisation rule. …