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If quantum mechanics is probabilistic, there is no reason for a particle to be in one place and not the other, but particles do make up their minds... but how?

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In fact, for Quantum Mechanics Nature is intrinsically probabilistic: there is no law describing with probability $1$ a single event. In accordance with Copenhagen interpretation of the quantum formalism this must not be considered as a limitation to our knowledge, but it is simply the manner as things exist in Nature, it is an ontological claim. In this view, it is wrong to assume that there is some deterministic law regarding, for instance, the position and the momentum of a given quantum particle: These quantities do not exist simultaneously. A particle is not a small ball moving in agreement with Newton's laws. It is just the object described by the formalism of QM.

There have been several attempts to construct a somehow classical theory behind quantum mechanics (the most interesting one perhaps is Bohm's formulation of quantum mechanics), I am speaking about the so-called hidden variables formulations. It is however known that no such theory may exist completely classical in view of the experimental failure of the so-called Bell's inequalities. If anything somehow deterministic exists behind quantum mechanics it cannot be classic as it must satisfy a constraint called contextuality and it must violate the locality principle.

We could say that QM is deterministic concerning the law describing the time evolution of quantum systems just because Schroedinger equation admits a unique solution as soon as an initial condition is given. However, we cannot forget that we are speaking of a probabilistic state. The state at time $t$ in QM is nothing but the set of assignments of the probability of every outcome of every observable if one would perform a measurement of that observable at time $t$. Time evolution of probabilities is deterministic. This is the only deterministic part of QM.

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    $\begingroup$ It seems to me that the issue is indeed more complicated than a simple binary decision between determinism and indeterminism will allow and that one shouldn't try to trivialize is this way (even though I find myself guilty of doing just that constantly). What poses the greatest challenge in understanding to me personally is the number of degrees of freedom of quantum systems. I have not managed to gain a satisfactory intuitive understanding of the consequences of the tensor product structure of quantum mechanical state space. $\endgroup$ – CuriousOne Jan 17 '16 at 10:43
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    $\begingroup$ @DanielSank: Echo in the absence of noise is not pointless. It proves that these systems are reversible. If the underlying dynamics would be stochastic in any meaningful sense, the echo (or any attempt at phase conjugation) could principally not work, not even on pure states. That echos can somehow undo mixed states was not the point I was trying to make. $\endgroup$ – CuriousOne Jan 17 '16 at 10:51
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    $\begingroup$ @DanielSank: The measurement process isn't deterministic. As soon as we do a measurement we irreversibly change the system. In classical mechanics we don't have to think about that, in QM we do. $\endgroup$ – CuriousOne Jan 17 '16 at 19:40
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    $\begingroup$ @DanielSank: Because quantum mechanics is not just the measurement process. How do I take nothing but the axioms of probability theory and construct destructive interference with them? Einstein was right when he said that nature doesn't throw dice. One can not construct a hidden variable theory of quantum mechanical systems from nothing but random processes, either. $\endgroup$ – CuriousOne Jan 17 '16 at 21:37
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    $\begingroup$ @CuriousOne I don't follow your logic. I didn't say QM is just the measurement process, but you cannot possibly call something a theory of physics without telling me what I expect to see in experiment. QM has deterministically evolving amplitudes, which act as probability distributions when measurements happen. You can't seriously say that's a strictly "deterministic" theory. There are elements of both deterministic and stochastic evolution. $\endgroup$ – DanielSank Jan 17 '16 at 22:28
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Quantum mechanics is uncertain, but it's not probabilistic. There are some very important differences between the two terms.

In quantum mechanics nature limits our knowledge about which particular outcome a future measurement will have, but the possible outcomes of given initial conditions and their expectation values are perfectly deterministic (and causal) and they depend on both the initial state and the measurements we are performing on the system during its evolution. In stochastic systems the final state does not depend on what measurements we are imposing.

In stochastic systems all components describing a system's state can be measured independently with arbitrary precision and the measurement process does not change the state, but in a quantum mechanical system we can only measure the projection of the state that belongs to commuting operators independently and these measurements will change the state of the system irreversibly.

Most importantly, as long as we don't perform measurements along the way, we can evolve a quantum system in time and then reverse its evolution and get back to the precise initial conditions. Stochastic systems can not be reversed in their evolution this way, they will not return deterministically to the initial condition that we began with, no matter what we do to them.

Lastly, the term "particle" has a set meaning in classical mechanics: it is reserved for the case where we are studying the motion of a macroscopic body (like a ball or even a planet) and we decide to neglect all of its internal degrees of freedom (like its temperature, magnetization, electric charge, chemical composition, vibrations and even its rotation!) and limit our description to nothing but the movement of its center of mass. In other word, a "particle" does not stand for an infinitely small body, but it stands for the coarsest possible simplification of the problem of motion in Newtonian mechanics. It's not a type of physical object but a type of physical approximation.

When quantum mechanics is taught poorly, this approximation turns into the objectification of quantum mechanical systems as infinitely small objects, as if scale would be the deciding criterion about whether an object is quantum mechanical in nature or not. Nothing could be further from the truth. One can build superconducting magnets with masses of many tons that only function because they are in a macroscopic quantum state.

Even ordinary matter, at any size, only exists because of quantum mechanics. No classical theory can describe matter, at all. It all has to "magically" presuppose that matter exists, before it can describe its motion. In quantum mechanics the motion and the existence of matter can be described by one and the same theory.

Therefore one has to be very careful not to adopt this sloppy idea of "particles as being in some place". Particles in classical mechanics are in "one place" because we chose to ignore the details of the material object we are actually describing, not because there are some magical infinitely small balls in play. Indeed, if we apply what we really know about quantum mechanics and matter that is being restricted to a tight volume, what would happen if we tried to ever measure the position of a piece of matter with ever higher precision, it would create ever more matter in that volume! That is exactly what a particle accelerator does when it creates new "particles"!

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    $\begingroup$ "but the possible outcomes of given initial conditions and their expectation values are perfectly deterministic and they depend on both the initial state and the measurements we are performing on the system during its evolution." Ok but the individual realizations are probabilistic, so I don't get what you're saying. $\endgroup$ – DanielSank Jan 17 '16 at 6:45
  • $\begingroup$ @DanielSank: Individual measurements have uncertain outcomes, but one can do a spin echo even on an individual spin, so the evolution is not probabilistic. If the state of the spin would change randomly over time, such a reversal of the dynamics could not occur. Not sure I did a good job explaining that to the OP... agreed on that point. $\endgroup$ – CuriousOne Jan 17 '16 at 6:50
  • $\begingroup$ @DanielSank & CuriousOne : you are both non-realist but not with the same bag of reality(ies) ! it's why I like ph.se :) $\endgroup$ – user46925 Jan 17 '16 at 9:00
  • $\begingroup$ @igael: Quite the opposite, a realist sees the world the way it looks, experimentally, only the idealists are trying to make it look the way they would like it to be. In that sense a quantum mechanical "realist" is actually an idealist, besides being a person who hasn't fully digested classical mechanics, yet. :-) $\endgroup$ – CuriousOne Jan 17 '16 at 9:04
  • $\begingroup$ @igael and CuriousOne, I'm kinda lost. $\endgroup$ – DanielSank Jan 17 '16 at 9:07
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If quantum mechanics is probabilistic, there is no reason for a particle to be in one place and not the other, but particles do make up their minds... but how?

This topic is obscured by common confusions about what quantum mechanics entails. There are several different explanations of what is happening in reality, if anything, to bring about the outcomes of quantum mechanical experiments. One common idea that you have outlined is that somehow one of the possible outcomes of an experiment is chosen at random. If that were true, then there would be a problem with understanding causality.

But the random choice theory requires adding an assumption to quantum mechanics on top of quantum mechanical equations of motion like the Schrodinger equation. This additional assumption is sometimes called the collapse postulate. But the collapse postulate is entirely unnecessary. You can explain the results of experiments without it, see http://arxiv.org/abs/0707.2832.

So what does quantum mechanics without collapse say about causality. Consider an experiment of the sort that would usually be explained by a system choosing randomly among multiple possible outcomes: an atom undergoing radioactive decay. Over a single half life the atom changes as follows: $$ |U\rangle\to\tfrac{1}{\sqrt{2}}(|U\rangle+|D\rangle), $$ where $|U\rangle$ is the undecayed state and $|D\rangle$ is the decayed state. The probability is given by the square of the number in front of the states: $(\tfrac{1}{\sqrt{2}})^2=\tfrac{1}{2}$. After the half life, there are two versions of the atom. Before the decay it looks like there is just one instance of the atom, but this is misleading because you can write down the state of the atom as a sum: $$ |U\rangle=\tfrac{1}{3}|U\rangle+\tfrac{2}{3}|U\rangle, $$ and you can have similar sums with any numbers you like that add up to 1. So there are multiple instances of the atom before the decay, that can be divided up in various ways even though they are all identical and no experiment could tell them apart. After the evolution some of the atom's instances are different from others, but there is no fact of the matter about which version of the atom after the decay corresponds to a particular version of the atom before the decay. The deterministic rule relating the state of the atom before decay to the state after, just tells you what fraction of the instances of the atom ends up in one state or the other, it doesn't produce a correspondence between one instance of the atom before the decay and an instance of the atom after the decay. And this also means that although the evolution is deterministic it doesn't allow you to predict what outcome you will see because there is no single fact of the matter about which outcome will happen.

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