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The ghostly passage of one body through another is obviously out of the question if the continuum assumption were valid, but we know that at the micro, nano, pico levels (and beyond) this is not even remotely the case. My understanding is that the volume of the average atom actually occupied by matter is a vanishingly small fraction of the atom's volume as a whole. If this is the case, why can't matter simply pass through other matter? Are the atom's electrons so nearly omnipresent that they can simultaneously prevent collisions/intersections from all possible directions?

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Not a duplicate, but related: Protons and electrons occupying the same space –  rob Jul 14 at 20:12
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Basically it's because of the electromagnetic repulsion when the electrons of each body try to go through each other and also because of Pauli exclusion principle. –  Hakim Jul 14 at 22:44
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@BrysonS. The radius of for instance the hydrogen atom at its ground state is $a_0≈5.29×10^{−11}\rm m$. So if the electron was at the other side of the atom it will affect the magnitude of the repulsive EM force by only a tiny fraction that is insignificant. –  Hakim Jul 15 at 1:12
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@Hakim This is a good point, but remember that the gradient of the electric force is proportional to $1/r^3$, so it actually could be pretty sensitive to small perturbations for values of $r$ close to zero. –  Bryson S. Jul 15 at 1:16
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Because atoms are not very sociable and tend to keep their distance in crowds. –  Hot Licks Jul 16 at 11:32

2 Answers 2

up vote 104 down vote accepted

Things are not empty space. Our classical intuition fails at the quantum level.

Matter does not pass through other matter mainly due to the Pauli exclusion principle and due to the electromagnetic repulsion of the electrons. The closer you bring two atoms, i.e. the more the areas of non-zero expectation for their electrons overlap, the stronger will the repulsion due to the Pauli principle be, since it can never happen that two electrons possess exactly the same probability to be found in an extent of space.

The idea that atoms are mostly "empty space" is, from a quantum viewpoint, nonsense. The volume of an atom is filled by the wavefunctions of its electrons, or, from a QFT viewpoint, there is a localized excitation of the electron field in that region of space, which are both very different from the "empty" vacuum state.

The concept of empty space is actually quite tricky, since our intuition "Space is empty when there is no particle in it" differs from the formal "Empty space is the unexcited vacuum state of the theory" quite a lot. The space around the atom is definitely not in the vacuum state, it is filled with electron states. But if you go and look, chances are, you will find at least some "empty" space in the sense of "no particles during measurement". Yet you are not justified in saying that there is "mostly empty space" around the atom, since the electrons are not that sharply localized unless some interaction (like measurements) takes place that actually force them to. When not interacting, their states are "smeared out" over the atom in something sometimes called the electron cloud, where the cloud or orbital represents the probability of finding a particle in any given spot.

This weirdness is one of the reasons why quantum mechanics is so fundamentally different from classical mechanics - suddenly, a lot of the world becomes wholly different from what we are used to at our macroscopic level, and especially our intuitions about "empty space" and such fail us completely at microscopic levels.

Since it has been asked in the comments, I should probably say a few more words about the role of the exclusion principle:

First, as has been said, without the exclusion principle, the whole idea of chemistry collapses: All electrons fall to the lowest 1s orbital and stay there, there are no "outer" electrons, and the world as we know it would not work.

Second, consider the situation of two equally charged classical particles: If you only invest enough energy/work, you can bring them arbitrarily close. The Pauli exclusion principle prohibits this for the atoms - you might be able to push them a little bit into each other, but at some point, when the states of the electrons become too similar, it just won't go any further. When you hit that point, you have degenerate matter, a state of matter which is extremely difficult to compress, and where the exclusion principle is the sole reason for its incompressibility. This is not due to Coulomb repulsion, it is that that we also need to invest the energy to catapult the electrons into higher energy levels since the number of electrons in a volume of space increases under compression, while the number of available energy levels does not. (If you read the article, you will find that the electrons at some point will indeed prefer to combine with the protons and form neutrons, which then exhibit the same kind of behaviour. Then, again, you have something almost incompressible, until the pressure is high enough to break the neutrons down into quarks (that is merely theoretical). No one knows what happens when you increase the pressure on these quarks indefinitely, but we probably cannot know that anyway, since a black hole will form sooner or later)

Third, the kind of force you need to create such degenerate matter is extraordinarily high. Even metallic hydrogen, the probably simplest kind of such matter, has not been reliably produced in experiments. However, as Mark A has pointed out in the comments (and as is very briefly mentioned in the Wiki article, too), a very good model for the free electrons in a metal is that of a degenerate gas, so one could take metal as a room-temperature example of the importance of the Pauli principle.

So, in conclusion, one might say that at the levels of our everyday experience, it would probably enough to know about the Coulomb repulsion of the electrons (if you don't look at metals too closely). But without quantum mechanics, you would still wonder why these electrons do not simply go closer to their nuclei, i.e. reduce their orbital radius/drop to a lower energy state, and thus reduce the effective radius of the atom. Therefore, Coulomb repulsion already falls short at this scale to explain why matter seems "solid" at all - only the exlusion principle can explain why the electrons behave the way they do.

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@Bryson S.: I added some discussion of empty space, but I am not sure it is intellegible for someone without prior knowledge in quantum mechanics. However, the fact that matter does not slip through other matter is inherently quantum mechanical, and cannot be fully understood on the level of our intuitions. –  ACuriousMind Jul 14 at 20:27
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Electromagnetic repulsion AND the PEP. Might be nice to elaborate role the PEP is having here - what would matter be like with just coulomb repulsion? –  Keith Jul 15 at 6:46
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@Keith With just coulomb repulsion, you probably wouldn't have electron orbitals at all, since all the electrons of the atom would collapse to the lowest energy state (which is prohibited by the PEP). What effects that would have on chemistry is hard to imagine - I'd expect that either almost all the atoms would mostly behave the same way, or the exact opposite - the periodicity of the elemental characteristics would go away (since it's mostly based around the fact that (almost) only the outermost electrons participate in chemical interactions). Ignoring that, there would be little change. –  Luaan Jul 15 at 8:33
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@Bryson S: I am certainly no authority on this (and the Physics.SE is not the place to discuss this at length), but in my opinion, yes. It is rather dissatisfying to see the elegance of Newtonian mechanics - where there are indeed definite particles (well...planets) orbiting others in a well defined way, with a lot of free space between them - be so muddled because many curricula try at all costs to reduce quantum mechanics to classical pictures. The most profund insights of the last hundred years are QM and relativity, and neither is taught well at entry-level, since neither is "intuitive". –  ACuriousMind Jul 15 at 15:24
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Also of relevance here is the cross-section for the scattering, which indicates that in most cases the effective size of the particle for the interaction is not related to the physical size, if one calculates it classically as the OP suggests. –  auxsvr Jul 15 at 20:29

If it were possible for one object to pass through another object, then it would be possible for one part of an object to pass through a different part of the same object. Therefore the question asked here is equivalent to the question of why matter is stable. See this question on mathoverflow. That question was more about the stability of individual atoms, but in my answer there, I gave a reference to a paper by Lieb. Section II discusses the stability of bulk matter. The argument depends on both the properties of the electromagnetic interaction and on the Pauli exclusion principle. Therefore anyone who tells you that the stability of bulk matter is purely due to one or the other of these factors is wrong.

The same holds for the normal force. People will try to argue that it's only due to electromagnetic interactions or only due to the exclusion principle. That's wrong for the same reasons.

Although Lieb's treatment is complicated, there is a pretty straightforward argument that quantum mechanics is necessary for the stability of matter. There is a theorem called Earnshaw's theorem that says that a classical system of interacting charged particles can't have a stable, static equilibrium. It's not a difficult or deep result; it's just an application of Gauss's law. Letting the equilibrium be dynamical rather than static doesn't help, since then the charges would radiate.

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This was also a good answer, thanks for responding. –  Bryson S. Sep 18 at 13:59
    
Of charged particles interacting only electromagnetically, you meant? You can’t prove anything about stability or instability assuming another force/field that obeys an unspecified law. For example, it would be trivial to achieve a static equilibrium by compensating electrostatic attraction for a repulsive force vanishing faster than $r^{-2}$, or by compensating electrostatic repulsion for an attractive force that depends on distance weakly. –  Incnis Mrsi Nov 25 at 10:24

protected by Qmechanic Jul 16 at 18:17

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