Are a quantum mechanical system a chaotic (yet deterministic) system? The title is slightly misleading. I really want to know if the randomness and probabilities observed in quantum mechanics is really just the result of a chaotic (yet deterministic) system.
If it is not simply a chaotic system, how do we know?
 A: I'm guessing that when you talk about randomness you're thinking about the collapse of the wavefunction and that the the result of the collapse is apparently random. If so, most us currently believe that the randomness is only apparent and is the result of decoherence.
Decoherence describes the interaction of a quantum system with the environment around it. It's impossible to completely isolate any system from it's surroundings and that means the system must interact with the surroundings and be affected by them. For example if you measure the spin of an electron to find out if it's up or down, you and your equipment are interacting with the electron. Because the environment (i.e. you) has so many degrees of freedom the interaction looks random but it's actually deterministic and the randomness is just a reflection of our limited knowledge.
Whether quantum mechanics is an emergent theory that arises from some deeper theory is an open question. See for example Deterministic quantum mechanics on this site. There are differing views on this and both sides include well respected physicists. I don't think there is any way for bystanders like us to judge who is correct.
A: I want to examine whether the elucidated question:

the randomness and probabilities observed in quantum mechanics is really just the result of a chaotic (yet deterministic) system.

can be answered in the positive or the negative.
From the wikipedia entry one has an adequate definition of deterministic chaos :

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.

Do quantum mechanical solutions and expectation values fall within this definition?
1) 

rendering long term predictions impossible in general

does not describe quantum mechanical solutions, which do give long term predictions in a probabilistic format.
2)

these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.

Quantum mechanical solutions give probabilities of realization, and thus the future behavior is not fully determined.
So no, Quantum Mechanics does not fall within the province of mathematics known as deterministic chaos theory, with its strange attractors etc. 
A: 1* Below


*

*Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions. Some quantum systems are sensitive to initial conditions, like spin systems in a time-dependent magnetic field; a rough example. Some systems are chaotic, some are not. 

*Probability and what you call 'randomness' arise because we can not observe a system without changing it. We have probabilites when observing a system of observing multiple values; do not get this confused with chaos! 
Cheers!
A: Yes, you can interpret it as a chaotic, yet deterministic system. But with principally unknown initial conditions (of both observed system and the observer).
This interpretation is called Bohm mechanics.
