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Probability or Impossibility in QM?

  1. Aug 6, 2007 #1
    Let us say we find the position of an electron at time t to be here on earth. An instant before this, the electron must have been here on earth as well since nothing can travel faster then 670 million miles per hour. In other words the electron does have a speed limit, so 1 x 10^-15 seconds prior to locating the electron on earth, there would be no other place that the electron could have existed. The problem is that this instant before we discovered its position, quantum mechanic equations provide a probability that the same electron was on the dark side of the moon. Why does Quantum Mechanics provide us with a probability that seems to be an impossibility?
  2. jcsd
  3. Aug 6, 2007 #2
    The wavefunction that gives the location of the election may allow it to be anywhere, but the probability for it to be in most places (like the dark side of the moon) is vanishingly small. In your terms, that's "impossibility."

    By the way, do you mean "dark side" or "far side?" I only ask because Mark Twain made that mistake too.
  4. Aug 6, 2007 #3
    QM wrong ?

    Thank you for reiterating my point countryboy. QM seems to provide a probability, and I understand how small it is, that the electron would be found on the 'dark side' (literary expression) of the moon, yet general physics tells us this is impossible. Since it is impossible, yet QM does give a probability, doesn't this mean that one of the latter is incorrect???
  5. Aug 6, 2007 #4
    I guess the expression, literarly, would be "No."

    The "impossible" of classical physics is the limit of "improbable" in quantum mechanics. If QM were that easy, anyone could do it.
  6. Aug 6, 2007 #5
    I'm no expert on the subject, but to me it seems like you're arguing something like this:

    You solve the Schroedinger equation and find a solution for some particle which gives a non-zero probability to find this particle over some large region of space. You then perform a measurement at time [itex]t_0[/itex] and find the particle to be located and some position [itex]x_0[/itex]. You're then asking why the Schroedinger equation gives a non zero probability to find the particle at [itex]t_0 - t^{\prime}[/itex] at the coordinate [itex]x^{\prime} > x_0 + ct^{\prime}[/itex], which would apparently contradict other physical laws.

    The point is that if you had solved the Schroedinger equation with the condition [itex]\Psi(x,t=t_0)=\delta(x-x_0)[/itex] then you would not have found the original solution which tells you that the wavefunction is extended over some large region of space. I'm not sure if this make physical sense (because measurement is time irreversible, so solving the above for [itex]t < t_0[/itex] is probably unphysical somehow), but from a mathematical viewpoint it seems reasonable.

    Edit: The boundary condition should be for the probabiliy density not the wavefunction
    Last edited: Aug 6, 2007
  7. Aug 6, 2007 #6
    So if we agree, in the physical sense, that it would be impossible to find an electron at time t =1.00000000000000000000000001 seconds on the moon if at t = 1.000000000000000000000000 seconds it was located on earth, then why are QM equations considered valid if they say we do in fact have some probability that the electron will be found on the moon at t = 1.000000000000000000001 seconds? Someone please provide a concrete answer to this dilemma. I have yet to see one. Thank you.
  8. Aug 6, 2007 #7


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    But if we had measured the position of the particle at that earlier time and observed it to be on the dark/far side of the moon, that would have collapsed the wave function so that (assuming we're using a relativistic formulation of QM) it would give a zero probability of being here on Earth.

    I think the wave function should be regarded as predictive, not retrodictive. It tells you where you are likely to find the particle now or in the future (assuming nothing collapses the wave function in the meantime), but once you have measured the particle's position, the previous wave function should not be interpreted as giving you information about where it actually was previously.
  9. Aug 6, 2007 #8
    Thanks jtbell I understand what you mean about the wavefunction only being predictive. For example though, if we find that an electron is on earth at time t = 0, then the function will give some probability that the same electron will be at the moon at time t = 0.0000000000001 sec even though light could not reach the moon at that time. Any ideas?
  10. Aug 6, 2007 #9
    Does relativistic QM really make that prediction (or just the non-relativistic approximation)?

    In general, does it bother you if things that you used to consider impossible (such as having an octopus quantum tunnel onto your desk) are actually just improbable (and so vanishingly unlikely as that you would expect them never to occur during the length of the universe)?
  11. Aug 6, 2007 #10
    Hi Frogman,
    QM REALLY makes such a prediction and such a prediction is what Albert Einstien could not come to grips with. And as for things that bother me, such as a particle traveling faster then the speed of light, Yes things like this do bother me and I, along with any other knowledgeable person, know things like this aren't just improbable, they are impossible.

    Thanks Frogman.
  12. Aug 6, 2007 #11
    Just to clarify, by "non-relativistic approximation" I meant the Schrödinger equation. Can you give a reference supporting your assertion that there is an accepted relativistic QM prediction of non-zero probability for particles to be measured as moving faster than light-speed?
  13. Aug 6, 2007 #12
    Yes, this is a well-known paradox in relativistic quantum mechanics. I think it was first clearly formulated and popularized by Hegerfeldt. See, for example

    G. C. Hegerfeldt, "Instantaneous spreading and Einstein causality in quantum theory", Ann. Phys. (Leipzig), 7 (1998) 716; http://www.arxiv.org/quant-ph/9809030

    There are lots of discussions of this effect in the literature. However I haven't found one that would satisfy me. Here I would like to suggest an explanation wihich, I think, is acceptable.

    It is true that if a particle is prepared in a localized state at time t=0, then at any later time t it can be found with non-zero probability outside the sphere with radius ct. So, if a particle is prepared now on the Earth it can be found (with some probability) 10^-15 seconds later on the Moon.

    It is also true that if such a behavior was found with classical (not quantum) particles, it would mean a serious violation of causality. This is because one could find a moving reference frame in which the cause (preparation of the particle on the Earth) and the effect (detection of the particle on the Moon) would change their time order: the detection would appear earlier that the preparation. This is certainly not acceptable.

    However, here we are dealing with a quantum particle governed by probabilistic laws, so the causality paradox is not so obvious. Indeed, for the particle localized from the point of view of observer O its wavefunction is no longer localized from the point of view of the moving observer O'.

    F. Strocchi, "Relativistic quantum mechanics and field theory", Found. Phys., 34, (2004) 501; http://www.arxiv.org/hep-th/0401143

    So, observer O' will see that even at t=0 the particle can be everywhere (with some probability), even on the Moon. So, this observer cannot definitely say that the particle travels from the point of its detection (on the Moon) to the point of its preparation (on the Earth). So, there is no obvious violation of causality.

    This means that in relativistic quantum mechanics it is possible to have a superluminal propagation (with almost negligible probability), however this fact does not violate the law of causality, so it is not a reason for concern.

  14. Aug 6, 2007 #13
    Thanks for the input Eugene. I'll look into your idea further and try to convince myself that this paradox in QM is explainable because to me it seems like we must answer this before taking QM any further.
  15. Aug 6, 2007 #14
    Thanks Eugene, although this raises another question for me:

    Say my lab on the moon has big containers of many many confined particles. (Feels awkward to use photons, but electrons would suffice.) Let the particles from each container be labelled in some manner (say by spin up vs down, although it would also suffice if one container just held antielectrons to contrast with the electrons). Say also that my partner is near mars (and moving at rest with respect to me) with a very sensitive particle detector.

    At some event, I release one of the containers as a signal. Since that labelled subset of particles is now free, their wave-function will now extend "instantly" to my partner near mars. Because I released so very many particles at once, she will be able to detect the suddenly increased particle intensity (above the background) before a conventional light signal would have had time to reach her. Therefore she knows that the event has already occurred, and by her measuring the labels on the particles (even just checking whether she detected electrons or anti-electrons, and igniting an appropriately coloured flare in response), I will have communicated one classical "bit" of information to her, faster than light.

    Where above am I wrong? One thing that I can easily show is how this above kind of FTL communication (which in the context of SR is always "backward in time" for some observers) is sufficient to violate causality (we can set up a "paradox" whereby my accomplice wins their local lotto competition).
  16. Aug 6, 2007 #15
    This is where your (otherwise fine) example goes wrong. Your partner on Mars is moving. So, according to boost transformations of wavefunctions she will never perceive your particles as localized. According to her, the probability of finding your particles in her (Mars) neighborhood will be non-zero at all times. So, there will be no sudden increase in the "particle intensity" and no clearly distinguishable superluminal signal.


    ADDED: I anticipate your objection: "how come that a particle in a sealed container has a chance to be on Mars even for the moving observer?" I must stress that I was talking only about free particles which were prepared in a localized (delta function) state by the observer at rest. Particles in a container add another level of complication (interactions), and Hegerfeldt's theorem has not been proven for them.
    Last edited: Aug 7, 2007
  17. Aug 7, 2007 #16
    No. Recall I specifically chose that she is at rest with respect to me (which is why I only described her position as *near* Mars).

    I can get rid of the intuitive parts, and make it theoretically cleaner: Let us be always non-accelerating and at rest to one another, in Minkowski space-time, and well separated in space. No boost transformation, just a space-like translation. Rather than a "container", let there just be a large number of ideal non-interacting particles initially prepared in an infinite delta potential well (which I can switch off at any time, after which they are all free particles, and I can quickly collapse their wavefunctions myself if that makes you more confortable..).
  18. Aug 7, 2007 #17
    Hmmm.... It may be a little late to try to disprove QM.
  19. Aug 7, 2007 #18
    I am sorry, then I misunderstood your words "and moving at rest with respect to me."

    If your partner is not moving, then she will be able to detect the particles released by you shortly after their release. Yes, there will be superluminal propagation. Where is the promised controversy?

    Last edited: Aug 7, 2007
  20. Aug 7, 2007 #19
    Do you agree that there will be superluminal communication of classical information? (If not, why not? The controversy is because causality paradoxes follow trivially from FTL communication.)
  21. Aug 7, 2007 #20
    I agree that observer on Mars will measure a sudden increase in particle density almost simultaneously with your opening the container. Yes, this can be regarded as FTL transmission of information. Where is the (trivial, as you say) causality paradox?


    ADDED: I think I am beginning to understand your point. But I'll let you formulate it. I need to do more thinking.
    Last edited: Aug 7, 2007
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