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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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Order this book, take two weeks off work and enjoy: amazon.co.uk/Emperors-New-Mind-Concerning-Computers/dp/… – Killercam May 8 at 14:27
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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. – David H May 8 at 14:42
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Possible duplicate: physics.stackexchange.com/q/7/2451 – Qmechanic♦ May 8 at 15:00
   
"Not only does God play dice, but... he sometimes throws them where they cannot be seen." Stephen Hawking – Dan Neely May 8 at 17:28
That last comment reminds me of en.wikipedia.org/wiki/Black_hole_information_paradox ... – Abel Molina May 10 at 7:45
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3 Answers

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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 more modern topics in Quantum Mechanics (e.g. how can the rules of Classical Mechanics, which are deterministic, can be obtained from QM). 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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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? – Walkerneo May 8 at 16:21
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@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. – BlueRaja - Danny Pflughoeft May 8 at 19:14
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 – nervxxx May 18 at 7:54
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. – nervxxx May 18 at 7:58
Consider a system of radioactive atoms that decay. while decay rates can be calculated from QM, a reasonable classical model would simply be that each atom has some probability of decaying i.e. given by the half-life. I can't tell you which atom is going to decay precisely at what time, only that at some time later I can predict I will get roughly a certain number of radioactive atoms left. So it seems indeterminism also exists in the classical picture..? – nervxxx May 18 at 8:02
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The easy answer is "noone 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 space) so clearly something is wrong with it. It just happens to work real well for engineering.

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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. – Ben Crowell May 10 at 14:43
Fair enough. Anyway, Alex gave a much better answer so I should just have kept my keypad shut. – Groda.eu May 10 at 14:56
+1: I like the honesty of "noone knows" :-D in fact, if you modify the Schrödinger evolution like Ghirardi, Rimini and Weber did in their theory, you add a stochastic term to account for the indetermination of a measurement! The universe may act much more crazily than what Schrödinger equation suggests – Alex A May 10 at 16:32
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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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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). – Alex A May 10 at 16:45
Lastly, can you please elaborate the last part of your answer? I don't see how QM would be contradictory. – Alex A May 10 at 16:47
I'll check GRW theory thanks – agemO May 10 at 23:28

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