Is the universe fundamentally deterministic? I'm not sure if this is the right place to ask this question. I realise that this maybe a borderline philosophical question at this point in time, therefore feel free to close this question if you think that this is a duplicate or inappropriate for this forum. Anyway, I'm an electrical engineer and I have some basic knowledge of quantum mechanics. I know that Schrödinger's equation is deterministic. However, quantum mechanics is much deeper than that and I would like to learn more. If this question is not clearly answerable at this point than can anyone point out some recognized sources that try to answer this question. I would appreciate it if the source is scientific and more specifically, is related to quantum theory.
 A: To add to the other answers and lead to more self-study, I point you to papers about the Bell's Inequalities and the Free Will Theorem .
These two point to the fact that the observations of experiments we have already made of quantum systems are incompatible with a number of things you would like to believe, each of which is connected to the vague meanings of determinism.
In short, Bell's Inequalities force us to abandon at least one of the following to conceptions of the world:


*

*Realism, meaning that particles can be assigned a definite state  at any given time (also called hidden variables).

*Locality, meaning that information propagates at a maximum velocity (relativistic causality).


Thus if you want the universe to have a definite state at each point in time, you have to accept superluminal effects ("spooky action at a distance"), if you want relativistic causality, you will have to accept that the state of the universe is uncertain at some times. You cannot have both.
The free will theorem says something quite similar, and although, a bit more abstract, I think it provides a stronger logical basis for what I just discussed.
And now the questions:
If the state of the universe is not defined, what does it mean for it to be deterministic?
If effects preceded causes, would the universe be deterministic?
Definitely, do read the first article I linked, as it contains rock solid results which need some time and skill to be explained, but are accessible and will go a long way to help you in your quest!
A: You're right; the Schrödinger's equation induces a unitary time evolution, and it is deterministic. Indeterminism in Quantum Mechanics is given by another "evolution" that the wavefunction may experience: wavefunction collapse. This is the source of indeterminism in Quantum Mechanics, and is a mechanism that is still not well understood at a fundamental level (this is often called as "Measurement Problem").
If you want a book that talks about this kind of problems, I suggest you "Decoherence and the Appearance of a Classical World in Quantum Theory" by Joos, Zeh et al; it is a good book on this and other modern topics in Quantum Mechanics. It's understandable with some effort, assuming you know basic things about Hilbert Spaces and the basic mathematical tools of QM.
A: On a side note, if we live in a sufficiently large multiverse it may be that every possible state exists. The result is a super-determinism that looks like non-determinism within local areas.
A: Physics is not competent to provide an answer to this question, but we can approach it
in a reasonable way and point to evidence one way or the other.
The physics that we call 'fundamental' is, at the moment, general relativity and quantum field theory, and
combinations thereof. The equation of motion in such theories is deterministic. It is difficult
to propose non-deterministic equations of motion without thereby proposing unlikely things such
as faster-than-light signalling.  So this part of the evidence points to determinism.
Non-determinism is indicated by chaos theory in classical physics, by the unresolved
issues surrounding the quantum measurement problem, and by a third property which
I will come to in a moment.
In chaos theory trajectories diverge exponentially, but numbers in science are never precise. This
makes it debatable whether or not classical physics is truly deterministic, because the idea of
an exact real number in the physical world is sort of a fantasy. As soon as there is
a limit to precision, it can get magnified by this exponential sensitivity and soon come to
influence macroscopic phenomena. So, within the domain of classical physics, determinism relies
on a degree of precision in the physical quantities which may be an unphysical requirement
for all we know.
Thus, although chaos theory can be calculated theoretically on a deterministic model, what it shows is that the large-scale behaviour can also be consistent with a non-deterministic model in which the non-deterministic feature is tiny. It follows that all the empirical evidence is consistent with either view. So it is not true to say that the evidence points to determinism. The evidence here is neutral.
In quantum theory the measurement problem is unresolved. There is no consensus among experts that is sufficiently widely accepted to merit the claim that the issue is solved. Among the ways to solve it one can propose that the basic dynamics have a stochastic ingredient. Some aspects of the study of black holes hint at this, but it is an open question.
So far I have written enough to show that we do not know whether physical behaviour is deterministic.
Now I will put forward what I think is strong evidence that it is not. This is the behaviour of our
human bodies and brains which enables us to be reasonable---that is, to engage in
reasoned argument and come to understand mathematics and other things on
the basis of their reasonableness. If our brains were totally deterministic then we would think
the things we do because the motions of our atoms so dictated. It is hard to see what reasonableness
would have to do with it. This is not any kind of proof, but it is a suggestion that our
thought processes are able to be influenced by something other than a combination of
mere determinism and randomness. If so, then the basic physics of the world is of a kind which
would support this. This suggests the world is not deterministic. This, to me, is a reasonable conclusion and it is the one I draw.
Note, this is not about adding some mysterious extra dimension or field or anything like that. It is simply to admit that we have not understood the world in full and a more complete understanding will likely show us that such familiar things as electrons and quarks are following patterns as subtly yet profoundly different from quantum theory as quantum theory is from classical theory.
The connection between free will and the ability to understand is, I should add, not one which all philosophers accept, but it is a philosophically respectable position. Roger Penrose, among others, has supported it by means of a careful presentation which can be found, if I recall rightly, in his book Shadows of the Mind (this is independent of what he or anyone else may think about wavefunction collapse).
A: The easy answer is "no one knows". The Schrödinger equation is just an equation that old Erwin threw together that happened to fit the experimental data. It is not even consistent with relativity theory (second derivative of space but only first of time) so clearly something is wrong with it. It just happens to work real well for engineering.
A: I've never heard about a non deterministic theory in physics, classical physics is, quantum theory is (if I take the wave function of the universe its evolution is deterministic), general relativity is ...
And about the wave function collapse, it means that something not well understood happens when a system interact with another one which posses much more degree of freedom, it doesn't mean that something non deterministic happens.
Otherwise quantum mechanics would be self contradictory : if I take the wave function of the system {system I want to measure + rest of the universe} and use schrodinger the evolution will be deterministic, if I just take the subsystem {system I want to measure} and use the wave function collapse the evolution would seem undeterministic.
"Can you predict with certainty the result of, let's say, an energy measurement of a two-level system"
If I had the knowledge of the initial wave function of the universe and were able to calculate its evolution thanks to Schrodinger I would.
"Lastly, can you please elaborate the last part of your answer? I don't see how QM would be contradictory."
If I say "The collapse of the wave function means quantum theory is not deterministic" it would be contradictory with the fact that I can use Schrodinger on the whole system instead of using the collapse axiom and find a deterministic evolution.
A: Well i would agree with some that say "no one really knows".
The whole question about a totally deterministic universe has other connotations.
However i will throw another answer here.
i will use thermodynamic reasoning and a new look at randomness and meaninglessness (i had a blog post sometime ago about exactly this issue, the blog is closed but i will give a summary here).
For anyone who wishes to see the original post i have also posted it in a comment at  aleadeum.com
Summary:

If each effect is fully conditioned by it’s cause (without no
  variation or mutation whatsoever) then time could go back. Find this
  hard to understand? Just think of a smashed mirror. If the act of
  throwing the mirror to the ground could account for the whole effect
  of the smashed mirror, then it would have the SAME probability that
  the pieces could recollect (at some other instance) and come back
  together to the hand. This does not happen in this way. This means
  that some variation took place which is outside the scope of the cause
  (or causes) that led to it. In other words, of course there are causes
  and effects that are generated by them, but not fully conditioned. As
  such new information is generated which aquires meaning (as it is
  connected to the rest of the system, a-posteriori). So this
  synthesizes the naive determinist’s objection together with true
  randomness. The answer is that randomness is another facet of
  uniqueness and this is the meaning of true randomness. So to sum it up
  we have this:
RANDOMNESS = NEW INFORMATION = UNIQUENESS

An analysis of concepts of time in physics from Galileo to Newton to Clausius, Thompson and Prigogine along with the philosophical implications in philosophy and physics, which relates to the above argument can be found here "The Temporalization of Time, Basic Tendencies in Modern Debate on Time in Philosophy and Science", Mike Sandbothe.
