planck length doesn’t come up even once, it all boils down to these things: 1. electron has momentum, and from that follows it has a wavelength, and at the same time 2. orbit is stable, which means that after every “rotation” electron has to end up with the same phase, which means there is only a finite number of solutions to time-independent schroedinger equation for (hydrogen) atom (don’t bother solving it on paper for anything with more than one electron) and these things are spherical harmonics