The first type: Dimensional Analysis.
When a person is not moving, the wavelength of the (Speed is v~10¯¹¹ m/s) de Broglie wave caused by the thermal movement of molecules is approximately 10¯²³. If the person moves normally (v~1m/s), then the de Broglie wavelength is the Planck length of Lpl~10³³ m.
The wall penetration probability must decay exponentially with the strong thickness a, and at the same time increase with the increase in volatility. A very direct internal quantity that reflects the magnitude of volatility, which is the De Broglie wavelength. The greater the De Broglie wavelength, the greater the volatility, and the greater the probability should be. Therefore, the probability of a person crossing the wall should be as long as P=e﹣a/￡. For a 1 cm thick wall, the probability of a person passing through is approximately e﹣10¯³³.
The second type: The Simplest Formula.
Any standard textbook of quantum mechanics will give the «probability» of the electron passing through the barrier.
If you consider a person as a «particle» and make a rough hypothesis k~k, then the probability of a person passing through the wall is e-2ka. For a person k=2π/￡~10³³ m-1, the probability of passing through a 1 cm thick wall is approximately e-10³³, which is consistent with the result of the dimensional analysis. Here is a recommendation for the Cambridge University Magnitude Estimation Textbook (pdf), which describes the art of estimation in detail. If the deviation of each estimate in the estimation process is regarded as a random walk, then this error almost follows the 1/√n principle. In other words, the more estimates are used, the easier it is to balance the errors between estimates.
Therefore, I would like to state once again that I tried to calculate in a precise way but used some strange assumptions, which may not be more accurate than my rough estimate.
Supplement 1
So can this probability happen? Murphy’s law tells us that if something has the possibility of going bad, no matter how small the possibility is, it will always happen. Therefore, if you have the possibility of sinking into the ground without destroying the ground, you will definitely sink. But in fact, there has never been such a record, so the probability given by quantum mechanics is wrong, so quantum mechanics is wrong (just kidding…).
Let us make a serious estimate and see if a «wall-through event» can happen in the universe. The sun has about 10^57 protons, the Milky Way has about 10^11 suns, and there are about 10^11 Milky Way galaxies in the universe, so there are 10^79 protons in the universe. Let’s be an exaggeration, assuming that each proton represents a «person», then there are approximately individuals in the universe. Assuming that there is contact (or collision) between different macroscopic objects, for example, if people always stand on the ground, it is the contact between people and the ground, and the contact frequency is once per Planck time (that is, equivalent to assuming that time can be quantized, The smallest unit of time is Planck time), that is to say, each person tries to walk through the wall 10^44 times per second on average, 10^51 times per year. The age of the universe is 10^10 years, so the maximum number of times that all «people» in the entire universe have tried to cross the wall is 10^80+51+10=10^141 times. So in so many contact incidents, will the crossing happen? Obviously not, because this huge number of collisions is still too small compared to the probability of passing through the wall.
In fact, the total population that has existed in history is about 100 billion, 10^11 (this number seems a bit magical? How does it always appear), so the actual number of collisions is far smaller than just estimated with 10^80. Therefore, we can never really pass through a wall.
Supplement 2
Some people say that the probability is very small, is it possible? Yes, after all, the living Murphy’s Law is here.
Some people say that if the probability is 0, is it possible? Of course, there are. After all, the probability of drawing any number between 0 and 1 is 0, but you can always draw a certain number.
In a practical sense, is it possible to pass the wall with such a small probability? No. Even if you draw another bottle of the lottery every day during the 13.8 billion years of the evolution of the universe, the probability is not so low.