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Question about quantum tunneling?

  1. Jan 24, 2012 #1
    You know how every particle has a non zero probability of being anywhere, but what about outside the particles Hubble sphere? In other words, does is there exist a nonzero probability that a particle can travel at superluminal speeds across the universe?
     
  2. jcsd
  3. Jan 24, 2012 #2

    Simon Bridge

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    That is a good question.

    Let me set it up a tad more carefully... bear with me:
    When a particle shows up, it may find itself with a well-defined, but not exact, position and then zip off into space. Since it's position is known to some extent, it has a range of momenta (Heisenberg's Uncertainty) so it's wave-function is dispersive.
    dispersion of gaussian wavepacket

    The classical speed of the particle would be the group velocity of the wavepacket.
    But this picture means that there is a non-zero probability of the particle travelling a distance in less time than that implied by it's classical speed.
    More precisely: it may detected before it could get there at it's average speed.

    The question then becomes - could this get early enough that the detection implies FTL?

    The above wave-packet, being gaussian, does not actually have zero amplitude anywhere: it extends to infinity.
    This means that the above picture includes arbitrarily high momenta.
    More to the point, it means there is a small but non-zero probability that our detector goes off immediately!

    But the picture does not account for relativity.

    We could argue that since the probability of finding a (massive) particle with v = c is zero, then relativity should contribute to the shape of the wavepacket too.

    See:
    http://www.ece.rutgers.edu/~orfanidi/ewa/ch03.pdf
    ... section on causality and fig 3.2.1

    The pulse ends up having a signal front which cannot be faster than the speed of light.

    iirc. there were a few experiments set up to disprove this for elementary particles - to try to see if FTL detection could occur with carefully prepared wavefuctions... but they never came to anything.

    Anyway: back to your question...
    even though non-relativistic wave-mechanics gives you results that imply that a particle may have a non-zero chance of being anywhere in the Universe, this is not literally the case in real life. It is a model with approximations in it, and, importantly, the model does not allow for relativity.
     
    Last edited: Jan 24, 2012
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