This was a disappointing article. I found quite a few incorrect statements, and the overarching claim that "space inside an atom is full of clouds" isn't going to be particularly convincing considering that clouds are also.... mostly empty space.
Also no mention of the Geiger–Marsden experiments/Rutherford gold foil experiments that discovered the atomic nucleus and involve alpha particles passing through a gold foil and only being deflected with a low probability.
If you want to "bust" a myth, at least understand where it comes from instead of building up strawmen.
I am bothered by the constant talk of "virtual quanta." They are in a way abstractions as the top comment mentions, mainly because there are terms in the perturbation theory that "look like" photons and leptons that weren't there originally in the starting state. However, the "virtual" part of this is key, you can't actually observe them, so they might not really exist. They do predict changes to the interaction however.
The important detail here that drives home how virtual they are is they only exist then when the regime is perturbative, for QED, at large spatial scales, and for QCD, at small spatial scales, hence why you can talk about them in hadrons like protons and neutrons as the author does. In the nonpertubative regimes, for example, strong QED, you cannot use perturbation theory and there are no "virtual photons" you can use. For example, the most basic strong field QED, the hydrogen atom, has no virtual photons in the coulomb field, you just solve the hamiltonian with the q^2/r potential given to you a priori. As the fourier transform of ~ 1/r shows you, you would need an infinite number of photons to make a couloumb field. This problem was already known from classical physics, in which the energy density of a couloumb field including the divergence at the point charge is infinite.
I think what gives a kind of credence to the atomic void is that we see it at other scales as well.
For example, our solar system is mostly “empty” with the mass mainly concentrated in the Sun (and Jupiter). The galaxies are mostly empty as well, so that two galaxies can “collide” without a single star hitting each other.
So there is a certain intellectual pull to try to extend the analogy (wrongly?) down to the smallest level as well.
I guess the analogy still holds with gravity instead of charge? So while the "real" mass of each object is compressed so "small" that normally colliding will be improbable, their gravitational influence is much much bigger and pretty much fill the entire empty space. And this is without even considering the milky way center that influences the whole galaxy.
Interesting that galaxies can collide without a star hitting another star.. but would it also be that the distances are so vast that the stars dont even (or mostly dont even) affect each other?
I don't know about the percentage of affected stars in a typical galaxy collision but note that you don't need a collision for two stars to affect each other. We're not speaking of interaction like on a pool table. The gravity will significantly affect star systems that pass significantly close from each other.
> I don't know about the percentage of affected stars in a typical galaxy collision but note that you don't need a collision for two stars to affect each other.
True, but the way in which they affect each other is that they try their best to collide.
From that perspective, it's kind of weird if collisions don't happen.
Is it? Think of a single star during a galactic collision. It wants to collide with every star in the other galaxy. It gets pulled this way and that, so, in sum, it decides to collide with the oncoming galaxy as a whole. Unfortunately, despite having billions or trillions of stars, that other galaxy is mostly empty space. The poor original star that we were tracking was unable to collide with any star at all.
Well, why do we keep attempting to collide with the Sun instead of striking out for the center of the Milky Way? Everything you just wrote applies just as much to the Earth as it does to a star heading through a foreign galaxy.
Why would stars hit each other during galactic collisions when stars inside of a single galaxy already almost never hit each other? Apparently reading it here of all the stars in the galaxy, a stellar collision only happens once every 10 000 years.
Interestingly enough, when galaxies collide, the supermassive black holes at their centres do typically merge as their gravity is strong enough to attract each other.
It's by the way not particularly unlikely for stars to pass through other solar systems. Apparently about 70 000 years ago a binary star system flew through the Oort Cloud.
Thinking of the scales involved, even at the speed of light probably two masses couldn’t pull each other fast enough to collide (if you are colliding the two galaxies at the fastest manifestible speed which is the speed of the light).
That’s hand wavy because if you’re imagining like grabbing the two galaxies and smashing them together… that’s like giga levels of magnitude faster dynamics than the speed of light, ha!
You’d have to like put the galaxies on top of each other and leave them there, probably for years, for gravity to get everything moving quickly enough (and then for the speed of light to even bridge the distance!) before really anything would happen. I think.
An other aspect is that orbital mechanics are weird/chaotic. Specially when you get masses that are similar scale. Like why hasn't moon hit earth yet?
So with two galaxies colliding I would not expect the stars to hit each other as result of collision. But as result of interactions when speeds increase.
This is not at all what happened. \Rutherford, Geiger, and Marsden ran an experiment. The data was totally incompatible with a solid blob of atomic mass, but instead suggested that the positive charge of the atom was concentrated into a tiny space.
Yes but why is positive all we care about when it comes to the definition of space? If we fire neutrinos, basically 100% of the atom is 'empty space'. If we fire protons, 99% is empty. If we fire electrons, none is empty.
Why do we say it is mostly empty just because of 2/3 of these things don't interact? Disproving the plum-pudding model was significant but saying the atom is 99% empty space is misleading. We know there are billions of virtual particles appearing and disappearing all the time within the atom and space, they just cancel out.
The word “empty” carries no semantic content under this construction of the term. Terms like “empty” are always used contextually, hence, why only a fool would correct their partner for mentioning the empty refrigerator by remarking on the presence of air.
Unless you think Sagan would have been surprised by the presence of an electric field (mediated by virtual particles…) between electrons and protons in an atom it’s quite likely you’re choosing an obtuse understanding of the term.
This is kind of my point, Sagan was a great science communicator but really missed the mark here.
A solid wood table is not mostly empty by any common contextual definition. Photons do not pass through it freely, your hand won't pass through it freely, and the electron clouds of the carbon atoms in the wood are physical in almost every common sense of the term. They very much push back on any other electron cloud that comes near enough and are generally the only part of the atom that does the interacting.
While it might be true that almost all the mass is concentrated in the center of the atom, that's not what people mean by empty. Houses aren't considered empty just because most of the mass is in the foundation.
Carl Sagan was just trying make some of these observations intelligible to children.
I can't take this article seriously if it doesn't engage with the discovery of the structure of the atom, and disregards the experiments where they shoot stuff through and it passes mostly unperturbed - because well its actually mostly empty.
It turns out that Schroedinger's equation predicts the results of the gold-foil experiment, too, and is better in some ways than the old "solar system" model of the atom.
If you dig really deep in the article - somewhere in the middle sounded by typical garbage like "We have yet to learn how to reconcile the dual wave-like and particle-like behaviour of matter" - the author actually indicates the issue he has with the apparently empty nature of atoms:
`The association between this mass concentration and the idea that atoms are empty stems from a flawed view that mass is the property of matter that fills a space.`
Whats really happening is that for certain kinds of interactions electrical forces were important, and for others nuclear forces were important. We understand the nuclear forces by proxy of mass concentrations, and reach reasonable conclusions like "the atom is mostly empty".
I actually like the idea of visualizing an atom as a cloud in addition to older visualization of orbiting “points”. But I kind of use cloud loosely in my mind and add energy field.
Although neither is technically true it helps convey they idea that everything atomically is in a state of flux.
So if you could stop time and glance inside a atom you may think to observe something closer to the empty space idea. However this situation is impossible and the representation is almost like a tesseract represented in 3d. Impossible to represent in a way we consciously operate.
But we live in a state where things are constantly in motion and thus the existing state of an atom can be kind of though of as a cloud or maybe even better thought as an energy field with predictable states.
There's a much better classical model of electron orbits as spherical fluid spinning shells or "orbitspheres" that have motion along all great circle routes.
I really enjoyed reading the article until this pop-up of “why we need your support” not only it blocked half my screen, but it also spoiled that attention I was giving to the article. These Jimmy Wales style of donations demand must be rethought, it’s too intrusive.
My approach is just to close the tab at that point. There are already more articles on the web and in print than I'll ever manage to read, and the content will probably be available elsewhere anyway.
Maybe it's a pipe dream, but my hope is that if enough people close browser tabs in reaction to rude behaviour, it'll start to show up on analytics.
Thanks, I don’t think I am willing to trade that pop-up inconvenience with my privacy offered to Googles by going through their cache. But yes I more than happy to use Archive.org. Did you have to copy the url, submitted it to archive.org to get the free access version? Or is there another secret flow I am missing?
You can use the wayback machine extension to quickly open the latest archived version. If it's not already archived, you can also quickly trigger the process.
So normally you could prefix the URL with archive.today/... (which is not related to archive.org, but a separate paywall-bypassing site). However in this case it appears the site is blocking archive.today, see: https://archive.today/https://aeon.co/essays/why-the-empty-a...
Well-written article on descriptions of electron (and importantly, nuclei to a lesser degree) behavior and state.
A pair of strawmen to knock down in the process, most notably in the title and opening paragraphs; the second half way through, re mass concentration. clouds of moving fluff as a lay description of electrons isn't too bad. I guess the key part is to emphasize charge in addition to mass.
Tangent: I'm surprised that, at least according to [this paper by Sebens](https://arxiv.org/abs/2105.11988), Shrodinger's model of the electron wave function corresponding to a charge density (probability aside) is controversial among physicists. I suspect most chemists doing DFT agree with it.
> Shrodinger's model of the electron wave function corresponding to a charge density (probability aside) is controversial among physicists
It's more than "controversial"; it was shown not to work way back in the 1920s. That's why Born's probability interpretation was adopted--it was the only one left standing.
> I suspect most chemists doing DFT agree with it.
If all you ever have to deal with is atoms and molecules, the Schrodinger charge density interpretation sort of works--it's at least a good enough heuristic for that domain.
But quantum mechanics gets applied to lots of other domains besides atoms and molecules. The charge density interpretation breaks down in those other domains, whereas the Born probability interpretation does not. And physicists who had to deal with those other domains figured that out, as noted above, back in the 1920s, which is why Schrodinger's charge density interpretation was discarded, at least as any kind of fundamental aspect of quantum theory.
DFT is like it is a cloud of density charge with some aditional weird rules.
Or in another words, you must sum many rules to calculate the energy, an one of them is easy to remember because it's the result you'd get from a cloud of density charge.
> Time, however, comes into play when a molecule collides with another one, triggering a chemical reaction. Then, a storm strikes. The quantum steadiness bursts when the sections of the electronic cloud pour from one molecule upon another.
If only my college chemistry class started here. I really enjoyed physics but for some reason never grokked chem. This might’ve changed all that.
I wouldn't jump to a hasty conclusion; you've not seen what their final exam would look like.
"Write 4,000 to 5000 word essay, on the follies of VSEPR (valence shell electron pair repulsion) theory. Highest scoring work will be published on aeon.co."
It’s too complicated to start from first principles to do any kind of chemistry. We even only have a semi empirical formula (SEMF) for estimating the masses of atomic nuclei
I have a physics degree but I never took chemistry (not even high school, I was too busy with AP Physics). It’d be nice to learn eventually… anyone have any textbook recs? College level is presumably fine
> Leave them to evolve by themselves without human interference, and they act like delocalised waves in the shape of continuous clouds. On the other hand, when we attempt to observe these systems, they appear to be localised particles
Debunking one myth by perpetuating another, it seems. Collapsing waves via observation doesn't mean you need to look at it as a human.
Reading this, I came to realise that my problems with imagining elementary particles don't even start with the famous wave/particle duality - I'm having trouble visualising the "wave" state on its own already.
I found the idea of visualizing particles as "clouds" somewhat helpful in that regard.
Overzealous highly theoretical interpretation that tries to pass itself off as an obvious truth and wrongly asserts that other interpretations are misconceptions. In other words, a bunch of bullshit.
The empty space interpretation is still covered at the university level and is largely regarded as a good interpretation, though, as is the case with science, one that is lacking enough correlation with observed phenomenon to be considered "the truth." And the same is taught about the cloud interpretation.
Neither of these ideas are meant to represent any truths about anything. These are useful tools in making mathematical predictions about the universe. And in that regard, they both work.
The collective scientific thought is slowly migrating from seeings atoms as billiard tables to seeing atoms as cups of tea, with a stable wave pattern due to vibration, and a vortex in the middle to represent the nucleus. What the tea is made of is not a concern yet, only the shape of its surface.
But, what about everything under the surface down to the bottom of the cup? What shape is the cup? Does the vortex reach to the bottom of the cup? If I could look down the vortex, would I see the bottom of the cup?
Physics is nowhere near experimenting with such things. Metaphysics has a rather interesting belief that electrons are the smallest 3d particles and look like inflow sinks, positrons are correspondingly outflow sinks, and the flow is made of much smaller 4d particles, that are, upon closer inspection, appear to be similarly made of combining tiny inflows and outflows of 5d particles and so on.
What do you mean by empty? Every point in the atom contains some of the smeared electron, it's just low density, less than 1 electron mass per electron radius.
>Time, however, comes into play when a molecule collides with another one, triggering a chemical reaction. Then, a storm strikes. The quantum steadiness bursts when the sections of the electronic cloud pour from one molecule upon another. The clouds mix, reshape, merge, and split. The nuclear clouds rearrange to accommodate themselves within the new electronic configuration, sometimes even migrating between molecules. For a fraction of a picosecond (10-12 seconds or a billionth of a millisecond), the tempest rages and reshapes the molecular landscape until stillness is restored in the newly formed compounds.
Cool description how bout an animation or better yet an interactive App.
When molecules have formed, and are in relatively static conditions, even then animating them is quite difficult. The wavefunction is a complex function of position and time, so you have to show 6+2 dimensions on a 2D screen.
During molecule formation, you need more than quantum mechanics. You need quantum field theory, to create an accurate picture of the dynamics. Quantum field theory has a lot more dimensions, and there is stuff like virtual particle creation and what not. It is not at all easy to show this on a 2D screen.
this aligns with goal of the article
"We build new mental images of the quantum world one step at a time, even under the risk of tripping up here and there."
Is it? Isn't the "empty atom myth" bassically the Bohr model of an atom, which was created by Niels Bohr.
The reason why it persists is the same reason Newton's theory of gravity exists. It is relatively simple to understand, and carries a lot of explanatory power.
Personally I think its straight from Rutherford himself. I think the article itself feels like a strawman since its arguing a position that probably isn't even that controversial in the physics community, but is also completely worthless information to the layman.
Yes, the Rutherford gold foil experiment was basically where we went from "It's a positive sphere with some electrons stuck in it" to "Oh, looks like there's a super dense positive center and the rest is mostly empty space".
The author of the article is welcome to compute the mass density of the atom's periphery to the nucleus, and return to speculating whether it's fair to argue that the atom is mostly empty space or not.
For myself, I'm happy to refer to ethereal probabalistic cobwebs as "mostly" empty space.
I was actually pretty interested to see the diagram in the article showing how delocalized the hydrogen nuclei are in NH3. I do have a science degree, and I’ve seen plenty of electron probability density renderings, but the extent to which protons “spread out” was surprising to me.
The probability thing is misleading though, it's really all about energy. Matter is just highly concentrated energy and just because the nucleus happens to be much more concentrated energy, does that make the energy of the electron cloud non-existent?
As a thought experiment: If you have a peach, would you declare the fleshy part non-existent once the pit achieves a certain mass density?
The actual scenario is this: "If you have a peach, would you declare the fleshy part non-existent if it had a certain probability of reallocating itself to a single spot once the pit achieves a certain mass density and if you touch it, there is only a small probability that you can even feel that the flesh is there?"
Electrons never reallocate themselves to a single spot. There's no evidence that collapse actually happens, that's just one weird interpretation (Copenhagen).
The entire electron cloud produces an electro-repulsive force that is very real. That is what you feel, there's no small probability about it. This idea that there's some small probability that you hand will go straight through the table is ridiculous.
When you touch a peach, some of your electrons/atoms actually do become part of the peach and vice-versa. If we somehow turned up the tunneling probability, your hand would not go through things, it would simply dissolve into things more readily.
This is a wildly pedantic point about the probabilistic nature of electron clouds.
When interpreting electrons as particles, the atom is mostly empty. The position of the electron is not known with certainty, but the number and mass of the electrons is and it is very small compared to the size of the entire atom. Volumetrically electrons are point particles and take up no space at all.
When interpreting electrons as quantum waves the term "empty" is largely meaningless.
> When interpreting electrons as particles, the atom is mostly empty
The Bohr planetary model, which I assume is what you're referring to, is equivalent to Newtonian mechanics. It's simple to understand and a good starting point for high schoolers. It's even still usable for certain types of calculations.
However, like Newtonian mechanics, at a deeper level and according to modern understanding, it's fundamentally wrong.
Instead we use the Schrodinger electron-cloud model. According to which the ~nucleus~ (EDIT I mean atom not nucleus) is not mostly empty.
It's not pedantic to discuss this, any more than it's pedantic to discuss why Newtonian mechanics is less accurate than relativity.
> Instead we use the Schrodinger electron-cloud model. According to which the nucleus is not mostly empty.
The electron cloud and the nucleus are different things. The electron cloud on it's own is an empty volume except where the electrons are at any given moment.
>The electron cloud on it's own is an empty volume except where the electrons are at any given moment.
The article specifically rejects this way of thinking and makes the point that the electrons are not in the cloud. The electrons are the cloud. Thinking of them as little balls whizzing about so fast that it seems cloud-like is wrong.
Except saying that the volume of space is filled is also incorrect. Multiple particle’s probability distributions may overlap without those particles interacting with each other. They don’t “fill” space.
It’s common to say they behave as a particle and a wave, but it’s equally valid to say they don’t behave as a particle or a wave. It’s a distinct phenomena and thinking in terms of large scale things you’re familiar with is simply misleading.
Saying it behaves as a particle is wildly misleading, we should really remove it from the curriculum. If electrons ever behaved as particles or even if the wave function 'collapsed' into particles, they would immediately crash into the nucleus and the universe as we know it would cease to exist.
Quantized fields is a much better term. Also collapse of the wave function never happens, only increased entanglement.
Treating an electron as a particle is the most natural way to understand the operation of devices like an electron synchrotron or a cathode ray tube, though. Not helpful for atoms or molecules though, I agree.
I'm not sure about even the synchrotron, most people have the very incorrect perception that a single electron is being circled around rather than a continuous beam. This is so far from reality where we're doing something like 600 million collisions / second that I question the value of the analogy entirely.
The concept of a single particle as a point charge that isn't really interacting with everything around it is similarly flawed. We learn from a very young age to conceptualize that as the building block of matter, and entanglement as something weird. Instead it's the other way around: a single particle with a pure (unentangled) wavefunction is pretty near statistically impossible.
It sounds like you might disagree that the room I'm in right now is filled with air or that my cup is full of tea.
This article is trying to help laypeople have better intuition about things from the quantum point of view. It's full of evocative prose. Arguing about the definition of the word "full" is completely tangential to the point.
More, I just think it’s misleading. Even the most basic introduction to physics should cover the idea that the air in the room has a pressure to it rather than the room being full.
The point of physics is to have a unified system where the same description applies to the widest possible number of phenomena. Calling atoms empty or full isn’t a great description but neutrino’s behavior for example lends itself to calling them empty.
And those clouds aren't in the 3d space, because if we consider an ekectron travelling from one galaxy to another, the electron-cloud will reach gakactic size before it interacts with something and then it will suddenly shrink into a tiny electron.
a probability field isn't really ever empty unless zero or near-zero is considered an empty space. which is kind of like a painting on which the some of the canvas is background
A probability field is not a geometric volume. The phrase "empty" is meaningless to apply to it and doesn't provide laymen with any useful intuition.
Fields may have high or low energy densities, and we might compare that to the height and frequency of waves on the ocean in explanation. But just as the ocean is violent or becalmed, not "full" or "empty", so are fields neither full or empty.
>When interpreting electrons as quantum waves the term "empty" is largely meaningless.
wouldn't it still be pretty meaningfully rephrased as 'stochastic density[0]' or similar?
I would presume that a nucleus, as a superposition if its respective waves, has a far higher stochastic density than the rest of an atom. Or rephrased, an atom is mostly [near] empty space.
> Or rephrased, an atom is mostly [near] empty space.
This is fair. But the point is that a model which is fundamentally stochastic is more valid than one where particles are continuously moving through some subset of mostly empty space.
wildy pedantic? I found it enlightening that without the properties of the quantum waves / particle clouds a molecule could not be. So it seems pretty crucial then to consider the perspective of particles as quantum waves. Which is not something I'm very familiar with. So I really enjoyed this article.
Now we know there can be less than void, which is negative energy. So what we humans interpret as void is just a threshold of the energy at any given point in space, and every point has a level of it.
"A molecule is a static object without any internal motion. The quantum clouds of all nuclei and electrons remain absolutely still for a molecule with a well-defined energy. Time is irrelevant."
Nah bro. Marie Curie gave up her life showing this isn't true. Maybe the author is confusing "static" with "stable"?
What did Marie Curie show about the internal dynamics of molecules? I thought that she was doing radioactivity?
I think that what he means that whatever vibrational or rotational internal states the molecule would have, the molecule is almost always in a superposition of all of its possible states (a "smudge" of all possible vibrations, so to say), and because the states are cyclic/vibrational, there is no time evolution.
Here - I'll explain it to you (and what atoms and particles and molecules etc... are). They're human abstractions. Particles don't behave like particles because the programmer sat down and thought extremely hard how to create a universe with the best amount of variation and dynamics and complexity using the least amount of resources (i.e. compute power). Instead of modeling individual point particles and their trajectories at each time instance at each point in space - it programmed the universe instead in terms of particle trajectories and their dynamics (i.e. the universe only computes probabilities and probabilistic amplitudes for quantized events). Humans think of this behavior as being modelled as some sort of united 'field' but just like anything else this is also an abstraction. The dynamics and edges are there to make it possible to perform further information compression which plays a vital component in terms of the universe's evolution (i.e. the universe is an infinitely recursive function and the life forms within it are vital in enabling it to perform recursive compression state space transitions and to optimize its own information exchange dynamics).
Not really. Everything in the human brain is actually an abstraction. The image or your 'vision' of what's 'out' there is just a computation done by the human brain. It's a 'holographic' 3-dimensional projection which only gives you enough information to survive evolved through many years of evolution. If you want to play that game, we shouldn't be having this discussion at all but we are having a discussion so it's not really coincidental. There is no mountain it's information all the way up and down.
Not to be mistaken with modern computing, digital computing, or Von Neumann computing; computation is simply having states which change in a dependant way.
It's not even required that the dependencies be linear in time or discrete, except in the narrow case of Turing-computability.
Not that big of a mountain, for a thing to exist it has to be implementable, and only computable things are implementable or else they would not “fit” into a universe.
And yet we know that most* functions aren't computable. Are we saying most of mathematics doesn't exist? If that's true then how can we talk about it?
I'm asking these questions rhetorically, but they're serious questions that need to be answered (or at least attempted) to maintain even a pretense of intellectual coherency.
You certainly can make the constructivist argument that only the rationals exist and real numbers and everything that builds on them is some kind of fever dream, but personally I've never seen any even remotely compelling exposition of that position. Maybe that's just a "me problem" though? I really don't know I find that this kind of metaphysic pushes up to and sometimes past my cognitive ability.
Constructivism seems obviously* true to me, and these Aleph arithmetics do in fact seem like a fever dream, but I’m not qualified enough to provide the exposition you’re asking for. To me real numbers exist, but only as approximation functions bounded by available memory and compute, and not as members of an uncountably infinite set in some platonic meta reality.
Check out the works and interviews of Joscha Bach if you haven’t already, he’s influenced my thinking on this quite a bit.
*obviously not in the dismissive sense, but in the sense as “we hold these truths to be self-evident”
We don’t even get to “the reals” before this issue crops up though. Irrational numbers were rigorously studied before infinity and the reals, as far back as 500 BC.
I’m not a computer scientist so I may have gap here, but demonstrating two quantities are incommensurable (showing no unit makes up two quantities m and n times, m and n being integers), does not seem like something possible to approximate empirically or computationally in many cases. The precision required may be one step beyond your current capacity.
Constructivism may be right. But I don’t have a good argument for why finished computation or empirical approximation (there’s always limits to measurement) is the be all end all. Unless we take them to be the final adjudicators, why shouldn’t there be incommensurable quantities? We need very strong arguments they provide the final say, but we know they have limits, their capacity/memory.
Computation isnt a human concept, it's as natural as mathematics. It's something to be discovered, it can only work in certain ways as defined by nature.
Sorry but I disagree. Dirac disagrees here as well. We discovered the positron from mathematics not the other way around. Many other discoveries have been made due to the application of mathematics (check out AlphaGo and ChatGPT).
Mathematics being a exceptionally useful human tool is compatible with mathematics not being "natural". Must hammers grow on trees?
It is also worth pointing out that the basic mathematics of deep learning are quite old and relatively simple. It was actually the technological advancement of being able to economically perform trillions of grade-school arithmetical operations per second that unlocked it, not some "mathematical discovery".
I'm sorry but I'm not sure what you mean by mathematics not being 'natural.' Einstein, Feynman, and many others used mathematics to give us an amazing exposition into the universe. Feynman in fact has stated that a 'particle' takes all possible 'path' computations into account for every possible 'particle trajectory' that exists and that the path of least energy (i.e. least information exchange) takes precedence. If you're telling me this isn't mathematical then you honestly don't know much about mathematics.
Frankly, it's clear that you haven't studied mathematics in any real depth.
Everything can be described (read: approximated, modeled) in mathematical terms; that's the whole point! That doesn't mean mathematical objects and processes must exist independently of those descriptions.
If I have one widget, and someone gives me a widget, I now have two widgets. If I then need to give someone one widget, then I will have one widget left. If someone else needs a widget, and so I break my widget in half and we share, I have half a widget left. Then five people turn up and give me three widgets each. I now have 15 and a half widgets.
The basic, most fundamental principles on which mathematics is based are surely natural and discoverable.
Other types of mathematics are born out of these principles.
To me this is exactly why mathematics is not natural. Counting natural numbers is not natural.
> one widget
And what's a widget? This implies there are "objects" and an object has its "boundry". Counting is completely human-defined. You can count earth as one and mars as two, or you can count solar system as one and an atom in another galaxy as two. It's a man-made system that help us think.
Or you can count particle by particle... well never mind, we're in a thread of an article about why particles are actually probability clouds :)
Why is my mug is one object, instead of two, or three, or 10^26 objects? Counting is very abtrary. Seeing a mug as a whole instead of a bunch of sub-atom waveforms[1] is a choice (that our brain hardware made for us).
Hm, so essentially your argument is that an object is a label that we put upon a quantity of stuff?
I'm not sure that I agree, but for the sake of argument, even if I accept that principle, you can still count how many of that quantity of stuff you have.
If I decide that a mug is made up of one part, then if I get a second mug, I have 2 mugs. If I instead say that a mug is made up of 10^26 objects, and I get another 10^26 objects, I'll have 20^26 objects.
It's easier to count 1 + 1 than 10^26 + 10^26, but there's no change in the fundamental principle of counting just because we don't agree on the number we have to count up to.
All of these are predictable and create natural mathematics through addition and subtraction (or multiplication/division, which are basically just repeated addition/subtraction).
> Seeing a mug as a whole instead of a bunch of sub-atom waveforms[1] is a choice [...]
Our brain is doing that because that bunch of subatomic waveforms have useful properties when considered together. They can hold quantities of other bunches of subatomic waveforms, for example, whereas a different collection of subatomic waveforms like my desk would not hold my coffee.
Your contention that there's no such thing as an object seems a bit solipsistic, and more of a philosophical question than a relevant or useful way of thinking about the universe as we experience it.
Because we don't have direct access to reality. Axioms exist as parts of models, which are human abstractions. Thus at some point to formally talk about reality we have to make rigorous assumptions about it.
That human beings are as good as we are at finding axioms that appear to correspond pretty well to reality is amazing to me. It's a really interesting philosophical question to ask why it is that we are.
On the contrary, it's the people who haven't done advanced studies of mathematics that tend to be confused/wowed by the popular mysticism surrounding it.
There is also a certain type that leans into that mysticism for personal gain, which IMHO is irresponsible and promotes the myth that mathematics is inaccessible.
It's not really a matter of mysticism. It's metaphysics. The question "why does Mathematics work so well to describe our world?" is entirely valid and interesting.
No they used the equivalent of a crazy, maybe infinitely, dense “bit” and all “programming” is things invented by the human brain to model this unknowable “objective” architecture. The universe isn’t actually a computer and even if it is it’s a computer so beyond us that computer science isn’t adequate to describe it.
Well...sort of. The last part is my hypothesis. The first part isn't. Particles don't have a trajectory (location and momentum) within spacetime in the normal way we think of them as having. This has been known for a very long time :)
This is a very dense article that could use a little unpacking. The author presents NH3 ammonia as an example but it's worth looking at two others, molecular O2 and methane CH4. Molecular orbital theory explains O2 very nicely but there's still an ongoing debate about whether to use Lewis dot structures or MO theory to describe methane (a very important chemical) to undergraduate chemistry students. (MO is better, it's far closer to experimental reality). There are some interesting papers out there on whether undergrads should be told methane is a Lewis-dot model or a MO model, IIRC.
Really, though, the problem here is that molecules are so complex that exact solutions fail and this brings us into the realm of numerical solutions, group theory symmetry arguments, and so on. If you find this subject interesting I suggest getting your hands on a copy of PW Atkins / RS Friedman "Molecular Quantum Mechanics" and perhaps also PCW Davies semi-pop-sci-but-with-way-too-many-equations (1979) "The forces of nature". (doesn't cover molecules though is great on background)
This article could be trying to cover too much ground, but perhaps thats OK? It's hard enough to apply QM to electrons and their interactions without trying to bring the nuclear interactions in at the same time (beta decay defies pop-sci demos, for example, IIRC it's two 4D matrices interacting with each other and somehow out pops a neutrino, I guess Pauli understood it).
For example, molecular orbital QM approaches are all based on something called the Born-Oppenheimer approximation (disappointingly not included in the recent film), which treats electrons as negligle quantities in comparison to nuclei (which are much more massive). So if we ask chatbot something like (1);
> "system: explain the Born-Oppenheimer approximation, which takes note of the great difference in the masses of electrons and nuclei. Because of this difference, the electrons can respond almost instantaneously to the displacement of nuclei. Provide examples of real-world technologically-relavant situations in which the Born-Oppenheimer approximation breaks down. User: a graduate student in a solid-state physics program. Assistant: This complicated topic is worth careful analysis..."
So the chatbot tells us, probably better than I could summarize it:
> "Because electrons are much lighter than nuclei, they can respond rapidly to changes in the positions of nuclei. In other words, when the nuclei move, the electronic configuration of the system adjusts almost instantaneously to accommodate these movements. This allows us to consider the electronic structure as adapting instantaneously to changes in nuclear positions. This separation of timescales makes it possible to treat the electronic and nuclear motions as somewhat independent, greatly simplifying the quantum mechanical calculations."
But since I once worked down the hall from these quantum dot enthusiasts, I know to ask a follow-up:
> "Titanium nanodots are a system where the B-O approximation breaks down due to relativistic effects, is this correct?"
chatbot: "In heavy elements like titanium, which has a relatively high atomic number, the velocities of its electrons can approach relativistic speeds, particularly in systems with high electron density or high atomic confinement, such as nanodots. When this happens, the usual non-relativistic treatment of electron motion (as assumed in the Born-Oppenheimer approximation) becomes inadequate, and relativistic corrections need to be taken into account. Relativistic effects can lead to several interesting phenomena..."
That said the overall thesis (of the source article) that there's a ton of stuff apparently going on inside of even simple atoms like carbon is certainly true. I think Feynman talked about this a few times, he said something like he couldn't understand how all that activity could be taking place in such a tiny space or words to that effect.
Anyone trying to understand this stuff... there are rabbit holes inside the rabbit holes, and it's all a very active field of current scientific investigation.
