Chaos can emerge whenever there is a feedback loop....for example when a hockey puck slides across the ice its speed depends on friction and the amount of friction depends on speed.
I could see a group of particles exhibing chaos with a speed / magnetism feedback loop, however, not a single particle's motion being predicted by shrodingers equation.
The main difference between chaos theory and mainstream physics is that choas uses discrete math to deal with feedback loops instead of differential equations. The solutions using differential calculus never give chaos...however using discrete math applied to the same math equation can give chaos. Becuase chaos relies on quantized math I think it embodies the philosophy of quantum