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According to quantum mechanics, there is fantastically (astonishingly, astronomically, infinitesimally, ridiculously etc.) small probability for a book on a table to quantum tunnel through the table.

Of course, such a claim is effectively untestable. However, it raises a question.

If one holds that it is real physical possibility.... then how is that different from someone who claims that human height follows a normal distribution and then holds that "it is possible a human will be born who will be 100m in height".

SchrodingersCat
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Iv Nik
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    These questions... Pop out every few weeks; exactly the same wording etc. In all cases, you received more or less the same answers. – Tobias Fünke Jan 02 '23 at 14:06
  • @TobiasFünke I don't have idea what you mean. Could you, please, contribute with your answer or at least address the problem by your comments? – Iv Nik Jan 02 '23 at 14:37

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Technically speaking, quantum mechanics is a probabilistic theory of physics. It does not dictate the future state of a system with complete certainty, rather it states the probability of it being in different states.

Now, in some cases, the probability is $1$, which signifies a sure event. And probability $0$ indicates an impossible event. This is how the mathematical tools get a physical interpretation.

If I now have to quantify that for the $2$ instances you mentioned above, a person having $100\,\mathrm{m}$ height is surely a possible event, since the hypothesis suggests that human height follows a normal distribution. But the number of random and unbiased samples that need to tested to observe one such result will be huge. If you argue that the total human population count since the last $1000$ years shows no such instance of a human being over $100\,\mathrm{m}$ high, and that number should be enough to yield an occurrence of $1$ or $2$ such people at least, as per the expectation values calculated from the proposed distribution, then the answer is: the hypothesis is wrong. Human height follows a different distribution, or is statistically probabilistic and random.

Similarly, the probability of an electron tunneling through a table is very low and depends on the thickness of the table. Assuming a thickness of $2\,\mathrm{cm}$ for the table, using results of the rectangular potential barrier problem, the transmission probability comes out to be $<0.001 \%$. Combine that with the number of atoms the book has, and you will have a sample size comparable to the number of atoms in the universe, which will make this an "extremely rare but mathematically still possible" event.

SchrodingersCat
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  • At least in the conventional Kolmogorov measure theoretic probability a probability of 1 event is not certain and a probability of 0 event is not impossible. Example: the probability of picking randomly a composite integer is 1 while picking a prime integer is zero (the density of prime numbers up to $N$ is $\approx \frac{lnN}{N}$). – hyportnex Jan 02 '23 at 13:43
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    Thank you for the answer, sir, but i am still struggling to understand what is the difference between the two instances. To put in my layman's terms: i don't believe that it is physically possible for a human to be 100m tall. In similar vein, why should I believe that there is astronomically small physical possibility for the book to quantum tunnel? What is the difference here? – Iv Nik Jan 02 '23 at 13:51
  • It seems to me that at some point we reach distinction between "mathematically possible" and "physically possible". And the most successful physical theory ever is incapable to make clear where precisely that point is. The math "works" astonishingly well but cannot even address physical questions like the mentioned above. – Iv Nik Jan 02 '23 at 13:59
  • @IvNik It is not about what you believe. Classical probability is a quantity that does have a physical sense to it. Probability of an outcome is defined as the number of cases favorable for that outcome to the total number of cases possible. So if the probability of observing a 100 m tall man is $10^-{30}$, just as an example, then this means that if you randomly choose $10^{30}$ people, then out of them, 1 might be 100 m high. That's the information that you get from probability. Similarly, you don't need to believe that there is an astronomically small probability of the book tunneling. – SchrodingersCat Jan 02 '23 at 14:54
  • @IvNik Mathematically, the probability can be calculated. If that number is, say for example, $10^{-10}$, then if you keep the book on the table $10^{10}$ times, then you will observe once the phenomenon of the book tunneling through the table. Now, this number of trials that you need to perform to observe this is extremely large, like I mentioned above, something like $10^{10}$ or around. That might take a time greater than the age of the universe as well!! So it is extremely difficult to observe this phenomenon to happen. This is where physicists interpret the situation as extremely rare. – SchrodingersCat Jan 02 '23 at 15:00
  • @IvNik It still remains mathematically possible, but physically it becomes an impossible event. – SchrodingersCat Jan 02 '23 at 15:01
  • @SchrodingersCat So, the possibility of the book to quantum tunnel through the table is "merely" mathematical artifact of the theory and not a real-life physical possibility with astronomically small probability? – Iv Nik Jan 02 '23 at 15:01
  • @IvNik It is physically possible, just like what mathematics says. But the experiment that you need to do to confirm that theory might be an enormous task, probably impossible for a human being with a finite life span. – SchrodingersCat Jan 02 '23 at 15:04
  • @SchrodingersCat "physically it becomes an impossible event" i do not agree with such claim. It is astronomically small probability, but it CAN happen. It CAN happen tomorrow or it CAN happen in a month. Of course, it is very likely that we will never observe it before the end of the Universe. Assuming we CAN calculate correctly how long the Universe will last. – Iv Nik Jan 02 '23 at 15:06
  • @IvNik To be precise, it's not the mathematical and real-life physical aspects that are going against each other. Rather it is the theoretical statement versus the experimental result. Theory has put its facts in front, but state-of-the-art experiments do not have enough infrastructure to prove that theory. For the "book" case only though. Quantum tunneling of electrons have already been observed. – SchrodingersCat Jan 02 '23 at 15:08
  • @SchrodingersCat That's why I call it in the body text of my question "an effectively untestable claim". – Iv Nik Jan 02 '23 at 15:10
  • @IvNik True. Not just that it CAN happen, it ACTUALLY WILL happen. But how is it of any interest to us if it takes an infinite time to materialise? That is where the physicist concept of infinity comes into play. If the timescale of an event is too large compared to the timescale we are interested in, then the event is considered to "infinitely long" compared to the desired timescale. – SchrodingersCat Jan 02 '23 at 15:11
  • @IvNik Exactly. It is an untestable claim. But just because we lack the infrastructure to perform a suitable experiment does not mean that the entirety of physics behind quantum tunnelling is wrong. – SchrodingersCat Jan 02 '23 at 15:13
  • @SchrodingersCat It is FINITE non-zero probability. We do not need infinity, which is a concept and not a number – Iv Nik Jan 02 '23 at 15:13
  • @SchrodingersCat Untestable does not mean wrong. Anyway, it could point to a sort of incompleteness. We can't be sure. – Iv Nik Jan 02 '23 at 15:15
  • @IvNik That is a mathematical definition of infinity. The physicist concept of infinity is different. For example, $100$ can be considered to be infinite compared to $10^{-20}$. Human beings live for $100$ years suppose. So if an event takes $10^{5}$ years to complete, a physicist will consider that event to be "infinitely" long. But this is just nomenclature. – SchrodingersCat Jan 02 '23 at 15:16
  • @SchrodingersCat In similar vein, it is physically possible for a human to be 100m tall, you hold such claim? – Iv Nik Jan 02 '23 at 15:17
  • @SchrodingersCat It is the mathematical entities that get physical interpretation. However, a finite non-zero probability is just that - finite non-zero probability. – Iv Nik Jan 02 '23 at 15:19
  • @IvNik Whether it is physically possible to have a 100m tall human is a question that has been answered in biology most probably. That's not a physics question. Even if you have a statistical distribution governing the heights of human, it might be biologically forbidden. – SchrodingersCat Jan 02 '23 at 17:20