quantum mechanics - Why does electron orbital circumference have to be in multiples of de Broglie wavelengths?

Electron orbit circumferences have to be in multiples of its de Broglie wavelength, but what do those 2 have in common?

Answer

I will assume you familiar with the properties of waves such as interference and diffraction.

Consider an electron orbiting the nucleus. By de Broglie's hypothesis, we would consider it to be a wave orbiting around the nucleus. Now, once the electron wave orbits once, the second time it would interfere with the first wave. For the system to be stable, that is, the electron shouldn't cancell itself out, so the wave must constructively interfere. That means, the circumference must be an integral multiple of the electron's wavelength.

Here's a picture that shows how the electron wave is expected to exist in the atom according to the Bohr's theory.