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From https://en.wikipedia.org/wiki/Quantum_tunnelling#Introduction_to_the_concept :

The reason for this difference comes from treating matter as having properties of waves and particles. One interpretation of this duality involves the Heisenberg uncertainty principle, which defines a limit on how precisely the position and the momentum of a particle can be simultaneously known.[7] This implies that no solutions have a probability of exactly zero (or one), though it may approach infinity. If, for example, the calculation for its position was taken as a probability of 1, its speed, would have to be infinity (an impossibility). Hence, the probability of a given particle's existence on the opposite side of an intervening barrier is non-zero, and such particles will appear on the 'other' (a semantically difficult word in this instance) side in proportion to this probability.

(Emphasis added.)

The bolded part doesn't make sense to me. I don't see how a probability can meaningfully be said to approach anything above one, let alone infinity. And regarding the latter bolded sentence, Heisenberg's uncertainty principle describes knowledge of position and momentum (not velocity) as complementary, so before that (seemingly misplaced) last bolded comma, should that say "momentum", not "speed" (with infinite momentum implying an impossible speed of c)?

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    $\begingroup$ I really think you should re-read the bolded part. It's not implying that "a probability can meaningfully be said to approach anything above one, let alone infinity" $\endgroup$
    – Gert
    Dec 8, 2020 at 21:48
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    $\begingroup$ @Gert "This implies that no solutions have a probability of exactly zero (or one), though it may approach infinity." What may be approaching infinity, then? $\endgroup$
    – Jacob C.
    Dec 8, 2020 at 21:50
  • $\begingroup$ Nothing, simply put. $\endgroup$
    – Gert
    Dec 8, 2020 at 21:50
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    $\begingroup$ I find myself agreeing with Jacob that the wording does very strongly imply that probability is reaching infinity, so perhaps a small edit can be made to make the sentence clearer. $\endgroup$ Dec 8, 2020 at 21:52
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    $\begingroup$ @Gert Thus why it doesn't make sense. The word "it" appears to be referring back to "a probability". $\endgroup$
    – Jacob C.
    Dec 8, 2020 at 21:53

3 Answers 3

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Short answer: there is a major typo, the author was not trying to say that the probability may approach infinity, but instead that the probability may approach zero at infinity.

So what probability is the author talking about? It's the probability for the particle to be found in some specific finite region.

So the sentence

This implies that no solutions have a probability of exactly zero (or one), though it may approach infinity.

is trying to say:

This implies that no solutions have a probability of exactly zero (or one) for the particle to be found in any finite region, though it may approach zero for regions arbitrarily far away (at infinity).

This would accurately represent the meaning of the original sentence, but it would still have some problems. I would replace implies with might give us an intuition, and may with will.

As for the second bold sentence:

If, for example, the calculation for its position was taken as a probability of 1, its speed, would have to be infinity (an impossibility).

If I was editing the article I would probably just delete it, but if push came to shove, I would rephrase it like this:

We won't try to rigorously justify this intuition, but it might be instructive to consider what would happen if the particle had a probability of exactly 1 to be at some exact position. The position uncertainty would then be 0, which means the momentum uncertainty would be infinite - and that would in turn imply that the uncertainty in its kinetic energy would be infinite as well. But the kinetic energy, unlike momentum, is a non-negative quantity. However, if we have a non-negative quantity with an infinite uncertainty, its average value would have to be infinite! Therefore having a precise position would imply that the average kinetic energy is infinite, which is unphysical!

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The boldened sentence in your question is incorrect if we follow the sentence word to word. But, I think the writer did a bad job in putting forward his idea into words.

I think it can be reworded as follows. Though, I think this is a far from satisfactory answer to understand why there's a finite probability for the particle to reach the other side of the barrier. Also, it is very vague as to what the author meant by probability.

This implies that no solutions have a probability of measuring a physical observable of exactly zero (or one), though it's uncertainty may approach infinity. If, for example, the probability to measure the position of the particle at a given position is taken as 1, the uncertainty in the measurement of momentum would be infinity, and thus having a finite chance that the particle has an infinite momentum(an impossibility).

I would rather take a more abstract mathematical approach to understanding this situation. As the potential is finite, the wavefunction of the particle will only decay exponentially in the barrier, to follow the rules that the wavefunction along with its derivative should be continuous. Thus, if the width of the barrier is finite the amplitude of the wavefunction on the other side of the barrier is non-zero, and so is the probability of finding the particle on the other side.

To try to get a "more intuitive" understanding of tunneling seems futile to me because our intuition is mostly based on our real world mechanics which follow the Newtonian laws. I'd rather try to modify my intuition to accommodate the properties of particles with quantum mechanical behaviour.

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EDIT: this was a sloppy answer, well deserving of a downvote. I had the right idea, but I should've included the words "in any non-zero size region" - because of course the wavefunction can easily be zero at discrete points. I'll leave this bad answer as is, because I wrote a better answer (to redeem myself :-)).

They were trying to say that the wavefunction is nowhere exactly 0, though it may approach 0 at infinity.

Of course, fixing that sentence will still leave the passage not very well written.

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