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Probability in Quantum Mechanics

  1. Feb 24, 2007 #1
    I'm a biology student, but I take some interest in physics and have read 2 of Stephen Hawking's divulgation books about modern physics and often stop to read something about it in wikipedia. Wich has revealed to obviously not be enough to understand some characteristics of Quantum Mechanics.

    Even knowing the answers are on the article I'm reading on wipedia, I'm not able to understand it well, probably because they are written by physicists and therefore to some point for physicists. So I searched for some physics forum where maybe I could get some more friendly explanations. :P Can anyone help me?

    The main matter that annoys me about Quantum Mechanics and to many other people it seems, is the apparent randomicity or probability it brings, making the determinable universe I always liked to think of impossible. But I'm not able to understand where this probability comes from.

    I can understand that through the uncertainty principle, precise measures for both of a pair of characteristics for a particle or wave become unreachable for us, but that doesn't means the system itself doesn't have definite values, does it?

    I can understand the uncertainty principle through the observer effect, observing changing the observed, but it seems that's not quite the nature of the principle. I read somewhere you could get a measure observing some particle entangled to another, getting information on it without disturbing it.
    However, does a particle get entangled for position or momentum too, I've always seen examples for spin? Is there a pair for uncertainty also for spin?
    But if measures can be made undirectly, where does the uncertainty arise from? Can't one measure both position and momentum for it precisely undirectly like that, without uncertainty?

    Initially well defined states for wavefunctions start getting less precise over time, why is that? From wikipedia, it seems it is possible to measure else position or momentum quite exactly without getting the other measure. Does the lack of a measure in position influence the state of momentum over time?

    Wavefunctions describe reality through probability values over time. But why does reality HAVE to coincide with the wavefunctions, why can't they be merely an aproximation of our reality due to unknown variables?

    In wikipedia it says John Bell has theorized creating a quite reasonable theorem saying that if there were unknown variables, they should equally affect all particles, creating a correlation and therefore being detectable. But couldn't the variables have different states for different particles, creating different tendencies? Or maybe the same state on different particles being affected by similar but non equal conditions on the system?

    My main question would be why does quantum mechanics have to suppose a probabilistic universe. And why do physicists seek for so unnatural explanations to justify it in the different interpratations of quantum mechanics. Why can't quantum mechanics be just the model to describe a system and the wave function collapse be origined on its true nature revealed.

    I manage to see ways out of it in my personal view, but I realize they're due to my limited knowledge on it.

    Sorry for the large amount of (likely stupid) doubts I've brought. And also for any english mistakes/incoherences.

    I'll be really thankful if anyone can answer any of those questions.
    Last edited: Feb 24, 2007
  2. jcsd
  3. Feb 24, 2007 #2
    The following should clear up your question and hopefully answer some others.


    Feynman's two slit experiment

    Firstly you need to understand superposition, the above account of the two slit and the explanation should clear up a few things.

    Consider the principles of superposition and decoherence, that lie at the heart of the Copenhagen Interpretation this should clear up some of the issues you have about probability and also reveal why matter in essence appears undeterministic.

    A wave function is exactly what you say, a mathematical expression of something we cannot directly see, so it is not meant to be interpreted as a "snapshot" of reality, more as an approximation of what happens experimentally.

    The true nature of light waves are still a mystery, although we can infer it's properties from various experiments which attempt to measure where the wave isn't so as to leave the particle un-decohered. And of course the principals of superposition are aptly demonstrated by Feynmans two slit experiment, involving single photons.

    Another interesting point is how do we measure say a photon? We need to use something to detect it, by doing so though we not only destroy its coherence but also impart energy into the system, thus we can know it's position but not it's momentum as the detection has changed the particle/wave in question.

    I'm sure others are better able to answer some of your other questions, but hopefully the above links should resolve some of your issues.

    Also a very simple explanation of Bell's theorem, courtesy of Dr Chinese.



    A discussion of the EPR experiment and Bell's theorem.


    Once you have all these under your belt, it would also be a good idea to do a search on Particle wave duality etc, some of the discussions here should give you a good picture of the often confusing world of quantum mechanics.

    Some wryly humorous quotes from the Father of the Copenhagen Interpretation: Niels Bohr.

    Last edited: Feb 24, 2007
  4. Feb 24, 2007 #3
    I would suggest S.Gasiorowicz “Quantum Physics” instead. In addition, consider receiving degree(s) in physics; it seems, it is very difficult to do something in biology without it.

    In physics, the unjustified statements are not required to be confirmed experimentally by the obvious reason (it is impossible). Your statement is correct, therefore, it should be demonstrated unambiguously. It is not performed yet.
  5. Feb 24, 2007 #4


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    You're in good company. That's now already 80 years that nobody understands these issues perfectly !

    You have to know that at its barest level, quantum mechanics is a formal theory (that means, mathematics and some intuition you'll need to make the relationship between the mathematical inputs and outputs, and "things in the lab") for which you GIVE some kind of "initial state" (however, not as with classical mechanics ; rather, a specification of what kind of "result of measurement you are sure of), and which will spit out a probability distribution of what you are going to measure.

    You can stop there - many people do. The price to pay is that you have no clue of what's "really going on". This attitude is called the "shut up and calculate" approach.

    Or you can try to "understand" the stuff. And then you're in trouble, because that's what people have been trying to do for more than 80 years now, without conclusive success. That doesn't mean that no progress has been made on the question! But one thing is sure: there is no consensus, and the reason is that no story is entirely satisfactory from all PoV.

    You enter now the murky waters of "the interpretation of quantum theory", and "the measurement problem".

    Actually, it is not the "randomness" in itself that is giving problems. In fact, even though *in principle* classical mechanics is deterministic, the fact that it depends upon the real number system implies that *in practice* it is also stochastic, given that we can never ever specify initial conditions "correctly" (that is, by assigning one and unique real number to each coordinate). We will always have to specify an approximate initial condition, and chaotic systems then lead automatically to totally stochastic behaviour after some time.

    What is so disturbing in quantum theory is that fact that the randomness is not "ignorance randomness". It is not (as Einstein thought it had to be) so that it is our ignorance of a certain reality which makes that the outcomes can only be known stochastically (just as in the case of classical chaotic dynamics, only, with a fixed boundary). It seems to be some kind of "intrinsic randomness" which "pops up out of nothing".

    There is no difference in principle between the spin uncertainty principle (the 3 components of angular momentum are a bit like the position and momentum in this respect) and the position/momentum uncertainty principle. But spin is simply easier, both experimentally and theoretically (finite-dimensional spaces etc...)

    There's always something that will bite you !

    That's what Einstein also thought, that the stochastic character of quantum theory is like any other stochastic description, namely an "ignorance description". In other words, there's something missing in our description of reality, and that something determines what's actually going on. Quantum theory describes us the statistics of that something, but without giving us the details. Such an approach is called a "hidden variable" interpretation.
    It doesn't work. Well, it does work, but not as expected.

    There are 3 steps in this "discovery".

    The first one was a failure. von Neuman thought he had the proof, once and for all, that there is no "ignorance description" possible. He thought he had a mathematical proof of the impossibility of rendering the stochastic behaviour of quantum theory by a set of hidden variables. It turned out that he made a mistake in his proof.

    The second step was Bohm. Bohm devised a hidden variable theory, Bohmian mechanics, which is stochastically equivalent to quantum theory. (as such, his theory was a counter example to von Neumann's erroneous "theorem"). Bohmian mechanics is an interesting theory in its own right. It merits to be studied.

    However, Bohmian mechanics has a problem. It uses "action at a distance" (as in Newtonian gravity) in a very peculiar way: particles on Alpha Centauri can influence particles in my lab in just as much a way as particles nearby. This is a bit strange - but ok. However, what is really annoying with Bohmian mechanics is that the way that this "action at a distance" works, means that it cannot be a relativistically correct theory.
    (it can reproduce relativistically correct statistical predictions if this is put in by hand, but the inner workings do not respect the basic postulates of relativity).

    There's a lot of semantic wars over this, but the short statement is that Bohmian mechanics is not "local" (which is a necessary condition for a theory to be in agreement with the spirit of relativity).

    Now, this was encouraging. Maybe it would be possible to formulate a theory, similar to Bohmian mechanics, which IS local. Bell set himself out to try to find this. Unfortunately, he discovered himself that this is impossible. That is the content of Bell's theorem: no LOCAL hidden variable theory (in the spirit of Bohmian mechanics or other) can ever reproduce all the stochastical predictions of quantum theory.

    This is von Neumann's theorem, but with an extra assumption, namely, that the hidden variable theory respects relativity in its inner workings.


    That's where we are.

    Some people say that, if this is the case, then we should just kick relativity in the butt. They work on refinements of Bohmian mechanics. Others say that this is so ridiculous, that the statistical predictions of quantum theory must be wrong. Unfortunately, many experiments have been done that suggest that these predictions are correct. I am careful in the wordings, because there are some people who try to pick apart all of these experiments, and suggest that not everything is 100% sure (the detectors could work differently than expected etc...). Let's just say that the margin for this is small and decreasing.

    Other people try to make sense of quantum mechanics as it is. I could go into that, but I would seriously suggest that, if you're interested in this, that you FIRST learn quite well the formal side of quantum theory (and keep yourself happy with the "shut up and calculate" approach), because to understand the reasons and subtleties of each of these attempts, you'll need in any case to be fluent with the formalism.

    But just know that your question is legitimate, and that there simply is no easy answer you might have skimmed over.
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