A few days ago I was explaining Laplace’s Theorem to a friend of mine in a very shallow way (“There one’s was this guy who thought that if you’d measured everything and you knew all the laws of science, you could predict the things that would happen next”). Then I said that nowadays, this is generally accepted to be untrue with quantummechanics and all. Enter Mr. Know It All, who hijacks the discussion and starts to argue that Laplace could still be true, only not in the classical mechanical sense of the nineteenth century. He started talking about probabilistic events and quantum mechanical processes.
I quickly stopped the discussion that was going nowhere, but it *did * set me thinking: what is the Straight Dope on Laplace’s Theorem? Could it still the case that (according to science and physics) the whole world is predictable, or do quantum mechanical (and relativistic?) processes make this impossible in priciple?
I’ve heard that there is one way that relativity limits this. Even if you could measure all things perfectly using non-quantum physics, you could only measure events that were in your past light cone; that is, events whose distance from you in light-years of space was not greater than their distance from you in years of time. Thus, there will always be events happening now that you cannot know about, but that will affect future events.
I’m not sure the OP specificially mentioned any particular medium by which to measure the cosmos, but even supposing that you did use light, what prevents you from calculating everything that already happened starting with your oldest light data?
One of the reasons Laplace’s idea wouldn’t work - and neither would calculating on past data - was the subject of a recent thread, to wit: it is apparently not feasible to create a computer that would actually be able to completely calculate the whole universe (or even the whole world).
In fact I also remember a number of years back the old Laplace deamon thought experiment was proven wrong: to keep track of the necessary information, the deamon would have to use more energy than would be derived from him sorting out moving particles in two rooms.
Well no less than Werner Heisenberg of “The Heisenberg Uncertainty Principle” fame said:
“We cannot know as a matter of principle, the present in all its details.”
If you can’t know the present in all its details it’s going to be pretty difficult to predict the future with any certainty.
Your friend may say that we can still make predictions about the expectation values for ensembles of particles, but these predictions are no more than statistical guesses as to what is most likely to happen.
The universe, at its most basic, is probabilistic in nature and there’s nothing anyone can do about this. A quantum particle does not inherently possess both an exact position and momentum. And in QM there can be no such thing as a particle’s trajectory.
I don’t understand this. What do you mean by inherent? Or do you mean that no observer can measure those two parameters to accuracy greater than that dictated by the Uncertainty Principle?
To say that the limitation is the accuracy to which we can measure the quantity doesn’t do Quantum Theory justice. It’s true that we can’t measure those two parameters beyond a certain accuracy, but it’s not based on some man-made limitation of measuring ability – even if one had the techniques to measure to such accuracy, he or she couldn’t. In some sense, Reality is not “complete”.
This leads to an interesting philosophical assertion, basically that beyond this physical limit the quantities don’t exist. If we knew a particle’s momentum exactly, the uncertainty in its position would be infinite, at which point, you can’t really say the particle has a position, because it doesn’t have one position – it’s simultaneously everywhere, until we measure its position directly, at which point we lose information about the momentum.
Another (perhaps simpler) example of this existence idea: Let’s say an object (how about a racquetball) can have two possible properties (in this example, let’s say color - red or green), it is simultaneously both until we actually observe it. You could extend this by allowing it to have an infinite number of colors, so the particle is simultaneously every color until we go in and measure it to be red, or blue, or whatever it happens to be.
I forget who first put this idea forward, probably Dirac or Feynman. I’d have to check my Quantum notes to be sure. I remember the argument being something along the lines of “If we can’t know the property (experimentally or theoretically), then how can we say it exists?” i.e. if we can’t know the color of the racquetball before measuring it, how can we say it has a color? The resolution is to say that it’s simultaneously all colors, and when we measure it we force it to collapse into one color.
I’m not suggesting that it’s a matter of technology. But, in theory, does this “feature” exist because it’s impossible for physical observation to be non-interactional (by virtue of the observer being part of the universe as well) or would the same limitations be applicable to a supernatural observer?
IOW, does the limitation appear simply because we are part of the system or do particles really occupy multiple positions simultaneously?
At a level which is somewhere in between the basic popular version (can’t know position and momentum at the same time) and our full modern understanding of quantum mechanics, quantum mechanics can be described as a wave theory in which physical systems evolve deterministically according to Schrödinger’s equation. Though I am unfamiliar with the details of Laplace’s theorem, I believe it does apply to quantum mechanics in this form. If you knew the wave equation describing a physical system, you could, in principle, calculate exactly what state it will be in at all future (and past) times. (This ignores the issue of measurement, which is where talk of nondeterminism comes in.)
The catch with this is that the wave equation encodes a hell of a lot more information about a system than just the position and momentum of its particles—far more information than even exists in classical theories. Even for the simplest possible system—a single particle moving in free space or in some kind of potential field—the wave function of that particle has a complex (in the sense of imaginary numbers) value at every point in space. There’s no possible way to measure the values this equation takes on, because any interaction with another particle will knock it into a completely different form. Only if we had a God’s-eye view of the universe in which we could observe reality without interacting with it might it be possible for us to predict “deterministically” the behavior of quantum systems, and that’s ignoring some philosophical complications about what such determinism would even mean.
Of course, when people do discuss the philosophical complications of quantum mechanics, such a God’s-eye view is typically assumed. Wouldn’t be any fun otherwise.
An ensemble of absolutely identical particles is prepared to be in the exact same quantum state and then run through the exact same measurement apparatus. Given this you’d think you’d get the exact same results. But you don’t. Instead you get a statistically predictable range of results.
I don’t think even God is capable of explaining this in Laplace’s Newtonian mechanistic universe.
In between measurements the only knowledge we have of quantum particles is the wavefunction and all the wavefunction gives us are probability amplitudes. We can localize position by superposing wave components with an infinite range of frequencies but since momentum is proportional to frequency any speculation as to what the particle’s momentum actually is is meaningless.
Quantum particles can’t simultaneously have both a position and momentum and in fact in a very real sense the particles don’t even exist in between measurements. According to some quantum interpretations we bring reality into existance by observing it.
It is impossible, even theoretically, to predict the exact moment at which an unstable atomic nuclei will decay. The best you can do is give a statistical model describing when it may decay.
I don’t think this is strictly true. One of the fundamentals of quantum mechanics is that things aren’t deterministic, they’re probabilistic. This is why Einstein and others couldn’t agree with quantum mechanics – they believed in a deterministic universe, hence “God doesn’t play dice…” I was going to try and come up with a good example, but Ring’s first example is clearer and nicer than anything i could come up with.
Basically, between measurements, the system doesn’t have one wavefunction, it’s in a superposition of all the possible wavefunctions available to it. It only collapses down to one wavefunction when we measure it. And after we stop measuring it, the system somehow returns to its superposition state. So to say “if we knew the wavefunction describing a system without measuring it” is putting forth an impossible situation, because it doesn’t exist in such a state.
I don’t like the “God’s-eye view” model because it implies that there is some sort of reality beneath the limits of the uncertainty principle, as if we would be able to predict things if only we knew some property of the system in between measurements. But experiments and theory point to the fact that between measurements, such a property doesn’t exist, so in my mind, even God would only be able to see the superposition of states that the particle is described by.
Then again, who am I to say what God can and can’t do? But it just feels to me like these arguments ‘water down’ quantum theory’s elegance, complexity, and philisophical mystery.
The other flaw in the deterministic view is that it is based on causality. As physicists, we usually strictly adhere to causality, but there have been experiments going on in which it appears that in the quantum regime, causality can be violated in rare, specific instances. The jury is still out on this, and researchers are trying to determine whether causality is really being violated, or whether there is something going on that we don’t yet understand.
I’m going to side with JasonFin here. Jargon ahead:
The Schrödinger equation is really just a first-order differential equation on a Hilbert space. What this means is that given an initial state (i.e. vector in the Hilbert space), the Schrödinger equation determines exactly what the state (vector) will be at any future time. As an example, if you tell me that you put Schrödinger’s cat into a box at time = 0 seconds, I can give you an exact mathematical description, at any subsequent time, of what the state of the cat is; it’ll exist in a superposition of being alive and dead, and I can tell you exactly what the coefficients of this superposition are. Probability only comes in when I try to measure the system, thereby projecting it onto either the “alive” state or the “dead” state.
As to the issue of whether any meaning can be assigned to something that cannot be observed — well, that’s more GD territory, but here’s my take on it: Quantum Mechanics is certainly describing something real about the world, otherwise it wouldn’t work so well. There are, in fact, specific states and measurements (the states are called eigenstates of the measurement) for which measuring a certain quantity doesn’t change the state; for example, if I measure the energy of an electron in the ground state of the hydrogen atom, I’ll get -13.6 eV. If I measure the same again ten minutes later, I’ll get -13.6 eV. No matter how many times I measure it, I’ll always get -13.6 eV, since the ground state of hydrogen is an energy eigenstate. The same thing will happen if I try to measure some component of the electron’s angular momentum — I’ll always get zero.
So the fact that the position measurements of, say, a particle going through a double-slit apparatus are distributed according to a known distribution doesn’t say that quantum mechanics is probabilistic on a fundamental level. Rather, it just says that the particle’s state isn’t an eigenstate of position, and the only reason that probabilities are arising is that we’re trying to impose a notion on the state that simply doesn’t make sense.
Forgot to mention one thing: Laplace’s conjecture said that a “sufficiently vast intelligence”, i.e. someone with perfect knowledge of the Universe, could predict it for the rest of time; there’s a distinction between this perfect being and us mere mortals, whose measurements (even classically) can never be perfectly precise. (Hmm, is that billiard ball at x = 0 or x = 0.00001?) All that the discovery of quantum mechanics does is to move this “perfect knowledge” onto the wavefunction — where, granted, we mere mortals can’t access the initial state, even in principle.
thanks for all the contributions. I guess Ring’s answer (both his posts actually) pretty much nailed the answer and the question on the head. I guess the position of JasonFin (and MikeS) is similar to my friends position.
What I want to know is whether it is possible, judging from inside “classical mechanics” and from inside Relativity/Quatum Mechanics/Maxwells Thermodynamics, to have perfect knowlegde of the present state of affairs and to “perfectly” predict future states. This is possible in principle (I think) in classical mechanics, if you are a sufficiently vast intelligence. The question is whether it is possible in R/QM/MT.
It seems that it depends on interpretation to a certain extent: is the waveform equation compatible with Laplace’s conjecture or not? Does this approach still constitute prediction in Laplacian sense or, for that matter, in any sense. As **MikeS ** mentioned, it comes close to GD, but I hope to keep it in GQ for a little longer.
While it’s true that if you prepare a state to be in an eigenstate of an observable (of a measuring apparatus) you’ll get an exact result, this by no means implies that QM is not intrinsically a stochastic theory. All it really means is that in the expansion of the wavefunction the coefficient of that basis vector and thus the probability equals one.
Very true but the only useful information you can get from the state is probabilistic.
But it just feels to me like these arguments ‘water down’ quantum theory’s elegance, complexity, and philisophical mystery.