Why do electron shells have set limits? (1999) | Hacker News Reader

Why do electron shells have set limits? (1999)

madsci.org

56 pointsxingyzt62 comments

I feel like this answer is "no, there is no reason for that to be the sequence you are seeing: it is an accident, as you can see from this careful analysis which demonstrates exactly why it is that sequence"... it is like saying "I noticed a bunch of random processes fall into a normal curve, is there a mathematical reason for that?" and saying "oh no, that is entirely on accident; here, let me show you the math for how a particular kind of random selection causes that exact distribution you are seeing, along with the equations that characterize if... as you can see, no math is involved here: this was just an accident" :/.

In this vein, I really enjoyed 1brown3blue breaking down why prime numbers form patterns when graphed in polar coordinates.

https://youtu.be/EK32jo7i5LQ

That is one of the most artistic expositions of mathematics I have ever seen.

3blue1brown is most likely the greatest math teacher alive. I’ve looked far and wide and found nobody who competes. Any HN’ers have suggestions for other math educators working on his level?

I think the intention was to say only that it’s “an accident” that they are double quadratic numbers, not that they have the exact values they have. Which is intended to say that the reason is different even if the result is the same.

Is it "an accident" that 1 + 3 + 5 + ... + (2n-1) = n^2 ?

I suppose the only way to get quadratic numbers "non-accidentally" would be to have the electrons stand in n×n square formation.

These are the numbers that ‘wrap’ around themselves to form squares.

It always seemed natural to me that the number of electrons would grow quadratically with the distance (principal atomic number), just like the surface area of a sphere grows quadratically with increasing radius.

I feel like the author of the comment calling it "by accident" would be extremely unhappy with number theory. In number theory all sorts of properties seem to crop up "by accident" but turn out to have incredible consequences and you realize that virtually every "accident" couldn't have gone any other way without destroying all of mathematics.

The electron shells themselves are in reality 3D objects that go all the way to the center, so it is a little strange after all that it increases with the surface area.

Is this true? I mean its a model. Aren't electron shells the quantum states of the electron fields?

It is true. The classical wavefunction of a ground state electron overlaps pretty heavily with the nucleus of an atom IIRC.

Only for s-orbitals. Anything with a quantum number l>0 has 0 probability at the nucleus.

Is it? I'm not a physicist but it seems like another natural infinitesimal, sort of like how a black hole's singularity is a point of infinite density whose infinity isn't a factor in calculating the total mass-energy of the black hole.

s-orbitals (l=0) have non-zero probability at the nucleus. Anything higher (p-, d-, f-orbitals) have 0 probability at the nucleus.

Forgive me for posting this idiotic question in advance since I have very little knowledge of physics, but I have often been confused about one thing recently. From what I could gather a little isn't the whole electrons revolve in orbits thing proved false and outdated like universe is made of ether?

I heard that even the iconic atom logo is all wrong since it is proved that electrons don't revolve around anything rather come in and out of existence and present themselves in probability clouds. If that is true then why do we still keep talking about shells and energy levels?

Like I said my question may sound like a troll but I assure you it's not. Just want to know what's what.

the orbitals aren't neat rings but there are definitely orbitals. they're shaped weirdly as they represent the space two electrons will probabilistically occupy. you can't pinpoint an electron and say it is precisely here in the orbital but there are fairly high odds that it's in the orbital.

There are orbitals[0], but they're not so nicely circular like in that "atomic" atom that dominated from the 1950's.

Each energy level has a different shape.

Also, they don't 'orbit' like planets. It's now a probability space[1]. i.e. You will _likely_ find electrons in this 'orbital' somewhere, but could be anywhere.... but you have to go looking for them. Once you've found them, they're instantly somewhere else in a completely unknown area of that orbital.

[0] https://en.wikipedia.org/wiki/Atomic_orbital [1] https://en.wikipedia.org/wiki/Atomic_orbital#Shapes_of_orbit...

> instantly somewhere else in a completely unknown area of that orbital.

Mr Einstein may have a few words to say about that.

He would probably say “you are right; the Heisenberg uncertainty principle means that you cannot know both the electrons position and momentum exactly due to quantum uncertainty”

I think the momentum is still limited by the speed of light.

So if you measured in succession fast enough, the particles would move continuously. So the probability density should be time-dependent.

Continuously, but always in a new unpredictable direction, i.e. Brownian motion.

And I suppose if you keep measuring them fast enough, so much disturbance will kick them off the atom, so they are no longer in the shell. The shell is what happens when the electrons are left alone. (Not completely sure about this part.)

With QFT (Quantum Field Theory)[1] there are no particles at all - just perturbances of the corresponding fundamental fields. Moreover, seems like QFT is the dominant explanation to date. At least at the current level of understanding and according the fad. There is a problem with it - it's incompatible with the general relativity, so unlikely is completely true. But it explained so many phenomena. Some people think it's not even fundamental, nor the space, time, fields themselves - only geometry/topology is - see, for example, the amplituhedron[2][3]. While it doesn't represent our complex 3+1 dimensional reality, it's certainly a new way to look at things. Time will tell.

[1] https://en.m.wikipedia.org/wiki/Quantum_field_theory

[2] https://en.m.wikipedia.org/wiki/Amplituhedron

[3] https://www.quantamagazine.org/physicists-discover-geometry-...

Unless I'm missing something, the Amplituhedron is geometry over an abstract space and is just a clever way of simplifying calculations. Also, its proponents haven't yet been able to apply it to interactions that occur in our 3+1 spacetime, so far they've only applied it to toy models of physics.

Clever algebra is not what the Universe "does", and you'll find that 99.9% of the geometry or topology topics in theoretical physics are over abstract, high-dimensional spaces that are a couple of steps removed from the ordinary goings on at the particle interaction level.

Yes, I mentioned it doesn't describe our 3+1 dimensional reality. It's more than just the algebra/geometry though - it kills the locality and unitarity principles. So it has quite a few implications on how we think about underlying physics.

I've seen a clever way to derive QM out of GR equations by making a single assumption: ds2exp(Cexp(-iwt)) = c2*dt2 - dx2 - dy2 - dz2. This extra exp() factor modulates the ds2 slightly, and when mixed with GR equations, it makes the time-dependent Schrodinger equation. Whether this math corresponds to any physics is a question, but if this idea is right, then the psi-function isn't some abstract "probabilities cloud", but rather a standing wave on top of the spacetime field governed by the GR equations.

AFAIK a standing wave requires the interference of two waves going in opposite direction, as in a resonant cavity. How does that work in a boundless space? Do we need some kind of time symmetry for this to work?

The relativistic effects create this standing wave, i.e. the particle distorts the spacetime as it's moving and the relativistic "glitch" mixed with the modulated ds2 creates the standing wave. IIRC, the idea was to apply the Lorentz moving frame transformation to the ds2 x exp() interval to derive some bits of QM. To derive the non time-dependent SE, some bits of GR were needed and to derive the time-dependent SE, more GR was needed.

Is this glitch the same as what causes an accelerating charged particle to emit photons?

I have no idea. The "glitch" I meant is the Lorentz coordinates transform.

The Schrodinger equation is non-relativistic so your program seems flawed.

SE is just a diff equation that approximates physics well. It's non relativistic in the same sense as the textbook wave equation is non relativistic. The "true SE" is likely way more complex, includes some bizarre math, but is compatible with GR and all relativistic effects.

The relativistic equation is called the Dirac equation, with QFT governing the radiation etc.

The schrodinger equation is just not lorentz covariant

You might have missed it in the article since the word is close but the term is 'atomic orbital', in part because it's not an 'orbit'. The wikipedia page linked in one of the sibling comments covers the basic ideas without going too far off the detail deep-end.

You are correct, the Bohr model has been obsolete for almost a century. It continues to get trotted out because the more accurate model is quite unapproachable without a lot of math and study.

When someone says "revolves around something" they are mentally projecting a whole sequences of measurements of the electron's position, one after another, that trace out a path. We are evolved to regard the path as a natural thing as opposed to focusing on the measurements.

But we can't do that for electrons orbiting an atom, since measuring them involves wacking them with photons or something that bounce off in some direction so we can see that deflection. So all we get to talk about is the result of a measurement.

When someone says "it's in this orbital," that's a statement about the values of energy and angular momentum a measurement will produce. What measurements actually produce those? Ones that involve absorbing or emitting photons, mostly, and we measure those by the energy of the photon absorbed or emitted, and those don't give us information about the position of the electron. You may think that we could put an electron into a very high orbit and then try to triangulate where it was when it emitted the photon based on the angle...but then you have to know where the atom is precisely, which means another measurement, which means some unknown momentum imparted, which means the energy spectrum of the photon we're trying to measure gets smeared...

Meanwhile, if we measure position precisely, then we impart some new momentum vector to the electron. What the momentum vector would have been measured to be before that wasn't known, and so we have no idea how we have kicked it and the orbital (energy and angular momentum) can have various values if we measure them afterwards even if they would have always had a single value beforehand.

I can't disagree with most of what you've said and it gives quite a classical intuition to the problem but it's still very classically grounded.

You're right, it is, despite my care to only speak of the result of measurements, not properties of the state we measure.

Is the accurate model that complex, though? An electron is a standing wave: one could show the first two standing waves on a drum and say these are the lowest two electron energy levels. Higher levels create more sophisticated waves, and non-flat potential adds more complexity, but the idea remains the same.

Yes, you are right, standing waves is the exact model that is used. All the measurement problems with e.g. location are just a consequences of that, some properties of a standing wave may be "smeared all over" the wave, like its location. And some interactions with the wave will change it. Quantized things are just standing waves of different order or "frequency" in a confined system.

You just understood 80% if quantum mechanics.

An electron isn't a standing wave, so apparently it is that complex. :(

What is it then?

It's own thing, and thinking of it as a classical wave or a classical particle isn't accurate. You go through the math and the experiments and try to build and intuition for what it is in its own right without reference to classical analogies.

Electrons occupying their orbital can be precisely measured in some aspects. Energy most prominently: An atom does have distinct energy levels for its electrons and an electron changing between two of those levels absorbs or emits the precise energy difference between those levels. Thats how emission and absorption lines come to be. Thats what is called quantized.

So if I see a certain emitted photon I can know precisely what the energy levels and angular momentum of the photon were before and after the emission. But I do not know its position or momentum.

This is such a classic physicist’s answer. It starts off promising, with a lot of context and detailed exposition which leads one to hope that perhaps it will be pulled together into an enlightening explanation at the end.

But when one reaches the end, one finds merely the truism “It is this way because that’s what you get when you solve the equations.”

(yes, I have spent a lot of time working with physicists.)

That's because it's really hard to explain physics to non-physicists without using mathematics. The clever part isn't classifying the orbitals, it's finding the Schrodinger equation in the first place.

We knew electrons had wavelike properties, but what equation did they obey? Even with hindsight it's still a difficult thing to derive - the relativistic answer is actually wrong, for example.

You reach the truism because the equations take years to learn - in engineering you can just take the result and run but in physics you have to be careful to understand what you're doing.

The Schrodinger equation for hydrogen isn't that difficult to solve (try it), but conveying that process in English is difficult.

My understanding of QM is very naive, but Schrodinger equation seems really simple to me. I could probably even explain it to a 5 year old. I'd show a few pictures of standing waves on a drum surface: the first two energy levels create simple shapes, higher levels create more sophisticated shapes. Then I'd say that an electron is such a wave. Then I'd introduce the idea of electric potential: I'd put a heavy thing in the center and say that this is how the atom's nucleus warps the electric field; and this changes the wave patterns somewhat, but the idea remains the same. Then I'd say that often multiple standing waves mix together and we get a quasipattern that oscillates between a few standing waves. That's really it.

You're right. The way to explain it intuitively is that at the quantum scale all particles are wavelike.

In order for electrons to achieve stability (such as within an atom), they must create a standing wave. Similar to the notes on a guitar string, each electron orbital is simply a harmonic.

The complication comes from the fact that atoms are 3d, so instead you need to use spherical harmonics, but it's the same principle, just standing waves in 3d space. These should look familiar if you've seen rendered images of electron clouds: https://en.wikipedia.org/wiki/Spherical_harmonics

It is quite surprising that this intuition is basically never taught in classrooms.

It actually is, well in university it is. Not sure about college.

Thank you. That makes much more sense to me. I found this[1] video with an efficient explanation and visualization.

[1]. https://youtu.be/Ziz7t1HHwBw

Not sure what this says about me, but before I looked at the year I thought this was about electron apps as command shells

I was thinking like wow electron has been around a lot longer than I thought!

Or from a different timeline where its 1999 and you can create electron apps by bundling them with ActiveX and Internet Explorer 6 ~shivers~

Yeah- it's not a pleasant thought. It's like some horrible episode of twilight zone made just for developers.

There's a series I've enjoyed recently: https://www.youtube.com/watch?v=zKbZeUvPnWI&list=PLpH1IDQEoE...

"How Small is it" by David Butler. Offers a nice summary of electrons and goes deeper into the Higgs field.

My two cents - after watching it - is it could all (i.e., wave/particle duality of electrons) be explained a lot more clearly if we just accept there is another dimension in space that electrons move into and out of, which we've not yet figured out how to observe.

Would explain dark matter / dark energy a lot too. The model seems so incredibly simple to me that I just don't get why more people haven't tried to build on it (am so busy with other stuff I doubt I'll have the time to figure out unexplained Cosmological or other phenomena that could be better explained by this model).

But it seems really straightforward this way. Electrons are small enough, independent enough to move out of and back into another magnitude of space that we can't measure. To us in 3-space it looks like they leave and come back (the idea of a particle being a "wave with itself" is less clear if you ask me). Everything is particles, it's just some particles have axis of motion in space that we don't know how to perceive yet.

One can think about it by analogy. Suppose we all lived in 2 space. If particles in 2 space where small enough to detach from our plane and pop up and down above and below the plane they would seem to appear and disappear.

It's the same thing with 3 -> 4 space. At some point people will realize that thinking about space as 3 space only is like thinking that the world is the center of the solar system. Our three dimensions are simply the closest / most defined / perceived ones to us. They're not even the "central" ones. They're the closest one's to our perception / how we've evolved.

If you get into the smaller spaces of things, there's less binding energy that constrains matter and more flexibility in moving between other dimensions. Likely there's a major 4th dimension that matter moves through / oscillates if you prefer since it's smaller particles that move the most (though cosmologically we see the effects in the large). And likely once you get deep in the 4th, there's a fifth, etc.

Again the analogy from "Flatland" / 2d space is you don't realize the freedom to move into 3 space until you separate yourself from the structure of what Flatland is made of at the sort of macro level. For us in 3 space the macro level is atoms. As you dive deeper inside, you're not bound by these constraints.

Gravity, everything can be modeled as particle forces if you just realize there's dimensions we don't perceive at the macro level yet (but which can be perceived indirectly by their effects on our 3 space - just as in time you can figure out what something moving in and out of a plane is sort of doing in 3 space).

Talk is cheap, what can I measure from your theory.

I wonder the same thing too! Seems like if one could explain some of these better though: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_p... with the model, then that'd be it (just like https://en.wikipedia.org/wiki/Tests_of_general_relativity) - that's all physics is at the end of the day. Better models.

And arguably there's value though in looking at the patterns in the history of science and being guided by them.

Many versions of string theory not only add a single additional dimension, but 11 dimensions.

One problem is defining the shape of those dimensions. Each shape would result in a different universe and different predictions.

True yeah. Seems like the simplest to start with would be one more dimension though. I don't know enough about string theory to know why they always add so many more dimensions. Seems a bit data-fit-oriented but maybe short-sighted. I mean it's likely, like the OP article linked here sort of indicates, there's some sort of symmetrical structure. So maybe an electron moves through one or two other dimensions. But yeah, maybe string theorists know stuff I don't.

How would you reply to the Kochen-Specker Theorem [0] (or its consequence, the Free Will Theorem)? The main problem with treating subatomic particles as tiny spheres, rather than as waves, is that we immediately have problems with entanglement and Bell's inequalities. I would be interested in hearing how you're going to handle that.

[0] https://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem

Thanks for the question and the link.

My sense is that being able to move through other dimensions (specifically a fourth spatial dimension) and into/out of our 3 space happens ‘superluminally’ since it’s more or less instant. The speed of a particle in 3 space has no dependence on its speed in the 4th dimension (by analogy with flatland - the speed at which a particle on the plane moves out of the plane).

No reason to assume the 4th dimension is constrained by the speed of light. I guess I don’t know enough about Bell’s inequalities and entanglement (yet! Will look forward to reading more) but I think that might help. A fourth spatial dimension is both local and nonlocal (it affects particles at all points in 3 space but isn’t constrained by 3 space). And it makes sense that it shows up only when we get down into the fabric of 3 space (subatomically) since...honestly am still trying to figure that out.

Is our 3 dimensional “flatland“ hurling through 4 space really fast? Are we suspended like with a magnet on both sides of our flatland, holding particles in place - except for leptons and photons which can oscillate above and below the plane?

Is the many worlds hypothesis true? Do we mostly move through 4 space on an arrow of time? And there’s a clear inertia there. But at the edges we can move through other 3 spaces? Why is there such inertia? How do we not be totally bound by it and move without its constraints?

> I just don't get why more people haven't tried to build on it

You should look at Kaluza-Klein theory, where electric charge is just momentum in a 5th spatial dimension of space-time.

Because those are the values specified in the simulator's .ini file?

I suppose they probably upgraded to a .yml file recently..

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