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According to this answer tunneling probability depends, among other things I don't know, on the length of the barrier.

Due to length contraction when going at relativistic speeds, it seems it should be theoretically possible to contract lengths arbitrarily by going at an arbitrarily high fraction of the speed of light. I suppose that means the probability of tunneling can get arbitrarily high if we just go fast enough.

At what point does the probability of a particle (say, a proton) tunneling through, say a planet or a star, significant. Let's say >10%?

Is this a case where quantum mechanics and special relativity still play nicely together, or not?

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    $\begingroup$ But a length contracted star has a much higher density... $\endgroup$ – ProfRob Feb 17 '20 at 15:55
  • $\begingroup$ @RobJeffries If density is an issue, then that would make a good answer, as I am then not knowing something vital. The follow-up question would be whether there is any change at all. $\endgroup$ – kutschkem Feb 17 '20 at 15:58
  • $\begingroup$ Realistically you'd need to be a particle with a tiny cross-section of interaction... similar to neutrinos, much smaller than that of photons. Photons behave diffusively within stars and they take ~Ma to reach the surface when created in the centre. $\endgroup$ – planetmaker Feb 17 '20 at 17:19
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    $\begingroup$ @planetmaker, assuming the star to act like a finite potential barrier, however high it might be, will the probability wave still show some significant tunneling? Given that the width of barrier can be made exceeding small and also the particles energy increases with increase in velocity? $\endgroup$ – NiRVANA Feb 18 '20 at 6:09
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For quantum tunnelling density and velocity is arbitrary. Tunnelling might be the wrong word as it involves a spherical "wave" starting at the particle extending to infinity, the Schrodinger wave. It is the probability of the particle existing at any point on the wave. The wave-function "collapses" when someone observes the particle, and the particle ends up at one place on the wave, almost always the point of highest likelihood, however it can also pop-up anywhere on the wave however most of the time on or near the point of highest likelihood. To cross something as massive as a star the probability would be probably (depending on the wave) approaching zero however not quite zero, MUCH lower than 10%. I only gave a brief explanation, I would recommend looking at other sources that can explain this topic much better than I can.

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