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.

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    $\begingroup$ Order this book, take two weeks off work and enjoy: amazon.co.uk/Emperors-New-Mind-Concerning-Computers/dp/… $\endgroup$
    – MoonKnight
    May 8, 2013 at 14:27
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    $\begingroup$ A subtle point about the TDSE: it is deterministic in the sense of differential equations, and the only thing it determines is the wave-function. If the wave-function itself is tantamount to reality, then quantum mechanics (and any quantum mechanical universe) can be said to be deterministic. If, on the other hand, the wave-function is merely a probability amplitude for classical state variables, then reality is stochastic. Deterministic randomness is not deterministic. $\endgroup$
    – David H
    May 8, 2013 at 14:42
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    $\begingroup$ "Not only does God play dice, but... he sometimes throws them where they cannot be seen." Stephen Hawking $\endgroup$ May 8, 2013 at 17:28
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    $\begingroup$ Some topics that you might find iteresting include the concepts of locality (information cannot travel faster than the speed of light), realism (things exist even when not observed), determinism, and their relation through Bell’s theorem. $\endgroup$
    – Ruud
    May 15, 2013 at 16:46
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    $\begingroup$ It is philosophical. You have introduced a conceptual dualism which was derived from the world which presents a logical disjunction and then asked which side of the disjunction the world represents. Both and neither. The answer invariably is that neither concept is adequate to describe it. Its far more subtle and the concepts are not actually mutually exclusive if they are to be modified to fit into the world. Its a problem of where do you stop the chain of reasoning. Physics is removal of non-deterministic notions. $\endgroup$ Nov 14, 2013 at 12:41

7 Answers 7


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.

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    $\begingroup$ Does this necessarily mean that the universe isn't deterministic though? Doesn't this just affect what we can determine based on what we can observe? $\endgroup$
    – mowwwalker
    May 8, 2013 at 16:21
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    $\begingroup$ @Walkerneo: According to the Copenhagen interpretation of QM, it does mean the universe is non-deterministic. There are other interpretations of QM which allow for determinism though. Which is the correct interpretation (if any)? Currently, no one knows. $\endgroup$ May 8, 2013 at 19:14
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    $\begingroup$ I would argue that the idea of wavefunction collapse is just a tool to sweep things under the rug. A wavefunction only appears to collapse if you fixate your attention to one subsytem of the full system. But a measurement necessarily involves entangling the measured system and the measuring system, and in the process simply spreads the coherence from the initial state over both systems. There is no loss of information, as the wavefunction collapse picture would seem to imply - it's just that it's quite hard to unentangle the two systems. theoretically it's possible by some sequence of unitary $\endgroup$
    – nervxxx
    May 18, 2013 at 7:54
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    $\begingroup$ transformations, but in practice it's hard. anyway, in quantum decoherence (which i assume is the topic of the book you listed), there is no need for the idea of wavefunction collapse at all. It is simply not physical. Thus indeterminism of QM should not be attributed to this non-physical process of wf collapse (kind of like how results in QFT should not depend on the non-physical regulator or cutoff). Instead, indeterminism of QM is simply because the theory is probabilistic in nature. And in fact, you don't really need to invoke QM to see this indeterminism. $\endgroup$
    – nervxxx
    May 18, 2013 at 7:58
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    $\begingroup$ Other good references for understanding what decoherence can and cannot say about the measurement problem: Schlosshauer, Zurek. (That second one was my advisor.) $\endgroup$ Nov 14, 2013 at 10:19

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.

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    $\begingroup$ It doesn't matter whether we talk about the Schrodinger equation or any other wave equation such as the Dirac equation. They all give deterministic evolution of the wavefunction. $\endgroup$
    – user4552
    May 10, 2013 at 14:43
  • $\begingroup$ Fair enough. Anyway, Alex gave a much better answer so I should just have kept my keypad shut. $\endgroup$
    – Groda.eu
    May 10, 2013 at 14:56
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    $\begingroup$ Schrödinger equation works just fine in relativity. It takes on the form $i\hbar \partial_t \Psi[\phi] = H \Psi[\phi]$, where $\Psi[\phi]$ is a wave-functional over the configuration space of field configurations $\phi$. It's very much fundamental to quantum mechanics, it seems. It's just not a field equation any more. $\endgroup$ Nov 15, 2013 at 0:21

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.

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    $\begingroup$ I disagree, for various reasons. First, how can you say that QM is deterministic? Can you predict with certainty the result of, let's say, an energy measurement of a two-level system in the state $\frac{1}{\sqrt{2}} (\left|0\right> + \left|1\right>)$? Also, if you haven't heard of non deterministic theories in physics, I suggest you to check out spontaneous collapse models, where the indeterminism is explicit in a stochastic extra term on Schrödinger equation (I'm referring mainly to the GRW theory). $\endgroup$
    – Alex A
    May 10, 2013 at 16:45
  • $\begingroup$ Lastly, can you please elaborate the last part of your answer? I don't see how QM would be contradictory. $\endgroup$
    – Alex A
    May 10, 2013 at 16:47
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    $\begingroup$ @AlexA the 'indeterminism' part only crops up when you fail to see yourself as part of the system. You are made of quantum mechanical interactions too. When you 'measure' you interact with the system you want information out of. I cannot predict my sensory experiences of interacting with a 2^-0.5 * (|0> + |1>) system. But I know it is obeys several mathematical laws below the bonnet. The Born probabilities are an open problem, probably to be solved with Evidential Decision Theory and Physics together. Not by sweeping it under the rug of "Collapse." $\endgroup$ Jun 7, 2013 at 21:48

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!


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.

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    $\begingroup$ Well yes, one can always make an appeal to the idea that everything we experience is part of a larger thing which is unlike everything we experience. But in what sense is our understanding advanced by such claims? What test would show that they are wrong if they are wrong? $\endgroup$ Jun 16, 2019 at 13:46

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).

  • $\begingroup$ I would suggest you have made a category mistake in your reasoning here. Read Dennett not Penrose for this stuff. $\endgroup$
    – isometry
    Jun 16, 2019 at 10:04
  • $\begingroup$ @BruceGreetham Thanks; could you say in a few words what kind of category mistake you mean? I have read Dennett's book on consciousness and have a low opinion of it because it merely asserts when it should argue. $\endgroup$ Jun 16, 2019 at 13:40
  • $\begingroup$ I can but only briefly as it is somewhat off topic for Physics SE: "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." Dennett's key point is that "reasonableness" is a description from the intentional stance which is an independent but compatible description to the physical stance. But if you don't buy into that whole approach I probably cant convince you. $\endgroup$
    – isometry
    Jun 16, 2019 at 15:36
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    $\begingroup$ @BruceGreetham Yes I find that position unconvincing. It relies on or appeals to a massive coincidence in which unlike categories align with one another for no convincing reason. I agree this is not the place for a lengthy discussion, but this much may at least help other visitors to this question on this site. $\endgroup$ Jun 16, 2019 at 17:57

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


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:


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.


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