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A Uncertainty paradox?

  1. May 5, 2017 #1
    Imagine a spatial frame of reference attached to a point-like particle. It has x=0 since it is at the origin and p=0 since it is at rest. Having definite position and momentum is normally considered a violation of the uncertainty principle. How would you resolve this paradox?

    1. Position frames and momentum frames are not the same. I.e. there is no such thing as a common "spatial" frame, because such a frame would imply a common eigenstate for both position and momentum representations in the frame indicated above.
    2. This special case is an exception.
    3. The uncertainty principle applies only to actual measurements.
    4. There is no such thing as a point particle.
    5. Other.

    I have my own resolution of this paradox, centered on (3), but the implications are many and complex and would probably be considered speculative (though I think they are obvious and focused on the distinction between translations and boosts and their unitary representations*) so I won't relate it here, but I'd love to hear what others think.

    *If anyone wants to know my resolution, pm me.
     
  2. jcsd
  3. May 5, 2017 #2
    I dont think its possible to choose such frame of referance. You cant set a point which point particle is rest. Its just not possible due to QM affect so I dont think its a paradox.
     
  4. May 5, 2017 #3

    DrClaude

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    This doesn't make sense. You define a frame such that the particle is at rest in that frame. It doesn't mean that the quantum particle is found at a single point in that frame.
     
  5. May 5, 2017 #4

    Demystifier

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    1. is correct.
    3. and 4. are correct in some interpretations of QM, but not all.
     
  6. May 5, 2017 #5
    I defined it as attached to the particle (at the origin).
     
  7. May 5, 2017 #6

    Nugatory

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    A frame of reference is a rule for assigning coordinates to events in spacetime; it's not "attached" to anything. We can perform a measurement on the particle, get some position out of that measurement, and declare that position at the moment of measurement to be the origin of our coordinate system. However, that doesn't attach the frame to the particle and there's no reason why a subsequent measurement of the particle should find it at the origin.
     
  8. May 5, 2017 #7
    If you don't like the word "attached", think of it as a location of the origin. An observer has a frame in which they are at the origin and at rest in that frame. That is what I mean by attached. But a spatial frame of reference, does not need an observer, it just needs a location and a path for that location. If we think of the particle as having its own frame of reference (as if it were its own "observer"), then it is both at the origin and at rest in that frame and we can think of the frame as moving with the particle so that the statements "at the origin" and "at rest" are time independent.
     
  9. May 5, 2017 #8
    The paradox is resolved by not breaking it in the first place. You began with "I have broken the uncertainty principle" and end with "now how does this reconcile with the uncertainty principle?". No such frame exists for electrons, which is the meaning of the principle.

    By which i mean, when you say "p=0" - p does not equal 0. I think the big issue missing here is how momenta and space are linked - TIME.
     
  10. May 5, 2017 #9
    So, you are saying that an electron (1) cannot have its own frame of reference and (2) that there is no transformation (translation, boost or acceleration) relating it to the frame of its observer?
     
  11. May 5, 2017 #10
    generally, yes. Take the double slit experiment: Which slit did the electron's frame of reference move through?
     
  12. May 5, 2017 #11
    Only the electron and the slit know that. There is no other observer unless you place one at the slit.
     
  13. May 5, 2017 #12
    I must correct you in that the slit and electron do not know either, hence the interference pattern (see 2:10 in vid).

    'Attatched to", "located at", "has a frame orginating from" etc. are different terminologies you are using for essentially the same thing. The point of unc princ is that you can't do any of those things if you want to have knowledge of electron momentum p no matter how you choose to paramaterise the spatial geometry.

     
    Last edited: May 5, 2017
  14. May 5, 2017 #13
    The interference pattern results from the projection of the prepared state onto the position basis of the screen. This takes place in the frame of the screen. The electron always knows exactly where it is in its own frame (i..e. at the origin). (Remember that a frame transformation is a unitary transformation and therefore can convert an eigenstate into a superposition or vice versa. So the unitary transformation representing the frame change from electron to screen will actually transform an eigenstate of position in the electron's frame into a superposition of positions in the screen's frame and therefore generating the observable interference pattern.)
     
  15. May 5, 2017 #14
    Apologies for the patronising childishness of the vid, i just wanted the pattern haha. I dont think thats correct but theres a lot of jargon to break through.
     
  16. May 5, 2017 #15
    Ben, I think your position is option (1) -- that there is no "spatial" fame as such and that position and momentum frames are distinct. This seems to be the popular position of others who have responded.

    My criticism of this position is as follows:

    Position and momentum eigenstates are related by unitary transformations. But so are frame transformations. If position representations and momentum representations both span the entire Hilbert space, this suggests that there is also a corresponding frame transformation that takes a position frame into a momentum frame. So what is this physically?
     
  17. May 5, 2017 #16
    I don't see the relevance of that math, i apologise. the way i see it is you're trying to model electrons as little flying axes, but whether you label it as a point particle or flying axes you come unstuck because the position IS undefined, not 0. This is because if I'm in a different frame, and want to transform to the electron frame, then I'm gonna need an operator that transforms [undefined] to [0] which i can't see having any nontrivial use? or maybe I'm mistaken

    Maybe you can assign a co moving reference frame centered on a quantum wave packet, but not a point electron at position x=R.
     
    Last edited: May 5, 2017
  18. May 5, 2017 #17
    No. You need an operator that transforms a superposition to an eigenstate. This is a unitary operator.
     
  19. May 5, 2017 #18

    Nugatory

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    "The frame of its observer" and "its own frame of reference" are common but sloppy ways of saying "a frame in which the electron is at rest" and "a frame in which the observer is at rest", respectively. It's easier to speak this way and the sloppiness generally doesn't matter, which is why we often use these not quite precise terms, but sometimes it can mislead - and this is one of those times.

    There is no frame in which the electron is at rest in the sense that repeated position measurements would the same result. There is a frame in which the expectation value of the second position measurement does not change with time and is equal to the originally measured position, and we could reasonably agree to call that frame "the electron's own frame of reference". However, in that frame there is uncertainty about the electron's position; the expectation value is constant, but a position measurement doesn't in general yield that result.
     
  20. May 5, 2017 #19
    Or "The electron is at the origin" and "the observer is at the origin". You can't specify which until you specify whether you are measuring position or momentum. In the absence of measurement, what is the distinction?
    Again this argument is one about measurement. It seems to me you are invoking option (3) and/or (4) but interpreting it as option (1) when they are not the same thing.

    I have no disagreement with option (3). I do however have doubts about option (1).
     
    Last edited: May 5, 2017
  21. May 5, 2017 #20
    Aharonov and Kaufherr (1984) covers this very topic
    https://journals.aps.org/prd/abstract/10.1103/PhysRevD.30.368

    They define quantum reference frames. They write,
    "The reference frames...can be thought of as laboratories containing rulers, clocks, etc., all of which are rigidly attached to the walls of the laboratory."
    The laboratories are finite mass, and therefore the laboratory obeys the uncertainty principle when viewed from another external reference frame.

    I haven't perused the paper yet, but they present certain paradoxes and claim to resolve them.
     
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