Question about quantum tunneling?

[zeromodz]

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?

 

[Simon Bridge]

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 traveling 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.