Why Can’t You Walk Through Walls? The Pauli Exclusion Principle and the Quantum Mechanics of Atoms

Image Citation: [1]

Written by Huyen Nguyen ‘28 

Edited by Leopold Li ‘28

Why can’t you walk through walls? The question looks like an easy one, and the answer is fairly straightforward. You are solid. Walls are solids. But things get a little more complicated when looking at it on a microscopic level. You could fit the entire human race in the volume of a sugar cube [2]. This statement is true, but how? Every human being is made up of millions and millions of atoms. Since all of these atoms have a lot of vacant space inside them and if we were to remove all that empty space from all of these atoms, the resultant mass of a human body would be infinitesimally small. It would be so small, in fact, that all of humanity would be equivalent to the size of a sugar cube. But, if the matter that makes up everything around us is mainly “empty space” (including the floors we walk on and the walls all around us) why can’t we walk through walls?

The reason that we can’t just pass through walls, the way that we imagine ghosts can, is due to a quantum effect called the Pauli exclusion principle. The Pauli exclusion principle states that no two fermions (a group of subatomic particles that includes electrons, protons, and neutrons) can be in the same state or same configuration at any one time, so no two electrons can exist in the same quantum state [3]. Spin does not mean that the electrons are physically spinning like little ballerinas inside an atom but it describes what the particle looks like if you rotate it in the role of angular momentum. In addition, the Stern-Gerlach experiment shows that the electron's spin is a quantum degree of freedom of the nature of angular momentum that carries a magnetic moment. It characterizes the electron's state independent of its position (or momentum) dependent wave function, or as observed, it is intrinsic (or inherent property of a particle) [4]. If a particle has spin 1, then it looks the same when it is turned by one full turn, an image which may help to explain this is an arrow–it must be rotated by a full turn in order for the arrow head to face the same direction as it did originally. A double headed arrow would face the same direction when rotated a half turn and so has spin 2. Electrons are peculiar and have spin 1/2—an electron only  looks the same when rotated 2 turns, which can seem counterintuitive, but all of quantum physics seems this way! This half spin makes electron fermions, which are the type of particles which obey the Pauli exclusion principle and therefore repel each other and cannot exist together in the same place at the same time [5].

Figure 1: [4]

So how does the Pauli exclusion principle prevent us from walking through walls? Well, electrons are found in discrete energy levels orbiting the nucleus. Each energy level is made up of orbitals, and in each orbital there is a maximum of 2 electrons, one with spin 1/2 and one with spin -1/2 [6]. Electrons like to fill orbitals before they start to pair up. Therefore the first electron in an orbital will have a spin of +1/2. After all the orbitals are half filled, the electrons start to pair up. This second electron in the orbital will have a spin of -1/2. If there are two electrons in the same orbital, it will spin in opposite directions. When our hand is pressed against a wall, all the electrons in our hand have to replace all the electrons in that part of the wall to be able to pass through it. And as you attempt to walk through the wall, your atoms ‘feel’ a repulsive force from the atoms in the wall, and their electrons are ‘squeezed’ together. Because electrons must obey the Pauli exclusion principle, they cannot be compressed into the same quantum state and must be in different shells due to the incompressibility of the volume, so your progress is halted [7]!

So, if the atoms align perfectly, is it possible to go through the wall? The space between the electrons and nucleus (or between electrons) is not empty, it is filled with an electric field almost a trillion times stronger than gravity [8]. Electrons, neutrons, and protons are not little balls; they are quanta—the smallest discrete unit (such as a particle) of a natural phenomenon in a system where the units are in the bound state. Electrons cannot come to rest, but even if they could, no amount of energy consistent with you continuing to have a hand could shove its electric fields through those of the wall—not by a very, very, very long shot.

Now, on the other hand, all this implies something called “quantum tunneling,” which in simple words means that since the location and energy of a quanta are probability functions, there is always a non-zero probability that when you measure it in some way, it will turn out to be on the other side of some seemingly impenetrable barrier [9]. Either it will randomly have attained the energy needed to cross it or will have randomly moved across it regardless. In practice, quantum tunneling has been observed over distances of less than 3 nanometers, that is, three billionths of a meter or a dozen or two atomic widths [10]. This is a very real phenomenon and is used in a kind of microscope, and it means that in fact, there is a very small chance that 3 seconds from now, that electron in your elbow might suddenly be an atom or two away from some other atom in your elbow, but very probably not. The fact is, the odds of a single quanta jumping all the way to the other side of a wall are so small, we can expect it never to happen in all the universe in many trillions of times the age of ours, and since the odds go  down exponentially with distance, if your hand did happen to hop through a wall on a quantum lark, it would almost certainly leave the rest of you behind since it is extremely unlikely to happen at any macroscopic level, following the Heisenberg Uncertainty Principle where it is impossible to know the position and momentum of a particle with perfect accuracy simultaneously. You are more likely to create a hole in the wall before you phase through it.

Figure 2: [10]

Let’s go back to the first paragraph: there’s a saying that atoms are almost entirely “empty space.” The idea that atoms are mostly empty space stems from their structure. Their nucleus occupies a very small fraction of the total volume, leaving the vast majority of space within the atom unoccupied by matter. Well, not quite, based on a more accurate description of atoms in quantum mechanics. In the context of the “flowing” electrons, the wavefunctions that embody the behavior of electrons are not like a solid “shell” around the nucleus, but rather they are a continuous, ‘space-filling’ function. Essentially, an atom contains no empty space; it is fully occupied by the wavefunctions. And these electrons exhibit both wave and particle properties, known as a wave-particle duality. We could look at them as made of waves, in which case they are not mostly empty, or we could look at them as particles, in which case they are made of points and have zero volume [11]. If you shove a bunch of electrons into a small space, it will take a lot of energy, even if you have a bunch of nuclei of opposite charge. The only way the atoms will line up to let your hand through is if they physically move out of the way and leave a hole big enough for your hand, which is unlikely but still vastly more likely than just quantum tunneling.

So, even if the electron orientations in your body and the wall were perfectly aligned, you would not be able to pass through because the Pauli exclusion principle prevents electrons from occupying the same quantum state, and the electromagnetic forces between atoms would prevent them from overlapping. And even if there is a chance that your hand can pass through, billions of atoms would have to do the same thing at the same time, which is astronomically unlikely, for you to safely travel your whole self through the wall [12].

References

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2. Ashish. Can All Of Mankind Fit Inside A Sugar Cube? Science ABC. Published September 22, 2015. https://www.scienceabc.com/pure-sciences/can-the-entire-human-race-fit-inside-a-sugar-cube.html 

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4. Quantum Theory of Electrons Continued. “Quantum Theory of Electrons Continued | Electronic Structure of Atoms.” Nigerian Scholars, 4 Jan. 2018, nigerianscholars.com/tutorials/electronic-structure-of-atoms/quantum-theory-electrons-continued/ 

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7. Boerner L. Why We Can’t Walk Through Walls. A Moment of Science - Indiana Public Media. Published 2015. Accessed March 31, 2025. https://indianapublicmedia.org/amomentofscience/walk-walls.php 

8. Siegel E. The idea that matter is mostly empty space is mostly wrong. Medium. Published April 26, 2024. https://medium.com/starts-with-a-bang/the-idea-that-matter-is-mostly-empty-space-is-mostly-wrong-540ef18819f7 

9. Flowers P, Theopold K, Langley R. Tunneling. Chemistry LibreTexts. Published October 2, 2013. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/02._Fundamental_Concepts_of_Quantum_Mechanics/Tunneling 

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