If like in Couder's picture we agree that QM is an effective model - describing our limited of knowledge, enhancing classical mechanics by the wave nature of particles (generated by some internal periodic process), we need to finally face the question of trajectories of particles - including electron's in atoms.

There are no doubts that such hypothetical trajectories of electrons should average to densities of quantum wavefunctions - and considering statistical physics on them: Boltzmann distribution among paths, it is clear that theromodynamically they should average to quantum densities (euclidean path integrals / Maximal Entropy Random Walk).

There is a nice experiment that literally makes photos of atoms: measure this electron density - by pulling out single electrons, measuring their position while leaving the atom and finally averaging over positions of thousands of electrons:

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.165404But the question is what are the short-time trajectories of electron in atom?

Standard answer is circular Bohr trajectories - which is still used especially for Rydberg atoms:

https://en.wikipedia.org/wiki/Rydberg_atomHowever, Bohr's model is known to be generally in disagreement with experiment - can it be repaired?

So Bohr model sees only charge of electron, while it has also extremely strong (for its mass) magnetic dipole moment - electron is also a tiny magnet.

This magnetic moment corresponds to "quantum" spin-orbit interaction, but it has also a classical consequence: Lorentz force for this tiny magnet traveling in electric field of the nucleus.

This force is perpendicular to velocity and spin direction and is proportional with v/r^3.

It is usually practically negligible, unless very large velocity and tiny distance - while free-falling, this force prevents electron falling into nucleus. Instead, the trajectory is bent and electron misses nucleus, then returns to the original distance, but on different angle.

This way zero angular momentum hydrogen can rotate - complex objects can rotate even having zero angular momentum, like in the falling cat problem:

https://en.wikipedia.org/wiki/Falling_cat_problemThis free-fall atomic model is claimed to give much better agreement:

https://en.wikipedia.org/wiki/Free-fall_atomic_modelHere is derivation of the classical spin-orbit correction:

https://dl.dropboxusercontent.com/u/12405967/freefall.pngSingle-electron simulator in Mathematica:

https://dl.dropboxusercontent.com/u/12405967/freefall.nbAnd somebody's simulation for up to 10 electron atoms ( youtube.com/watch?v=P2IsIkSn5bk ):