4D Chemistry

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

4D Chemistry

Postby Prashantkrishnan » Sun Oct 02, 2016 1:41 pm

I'm confused about the magnetic quantum numbers in 3D Chemistry. When we talk of p-orbitals, we generally name them px, py and pz. They are given the quantum numbers -1, 0 and 1. How is this justifiable, when x, y and z are interchangeable while -1, 0 and 1 aren't :?: :roll:

In 4D, I assume that there will be four p-orbitals. How will we assign magnetic quantum numbers for them?

Also, in 3D, there are 5 d-orbitals. Suddenly why do they start violating symmetry? (I don't know how they are obtained from Schrodinger equation)

The dxy, dyz and dzx orbitals seem alright. I fail to understand how the remaining two are dx2-y2 and dz2. And how come the quantum numbers -2, -1, 0, 1 and 2 correspond to these?

I assume that there will be 7 d-orbitals in 4D. What would they be, and what would their magnetic quantum numbers be?
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Re: 4D Chemistry

Postby Teragon » Sun Oct 09, 2016 7:27 pm

I've explained it a bit in the 2D chemistry section.

Prashantkrishnan wrote:Also, in 3D, there are 5 d-orbitals. Suddenly why do they start violating symmetry? (I don't know how they are obtained from Schrodinger equation)


Why should they all have the same symmetry? There are just more different possible standing waves with different shapes with the same orbital angular momentum.

Prashantkrishnan wrote:The dxy, dyz and dzx orbitals seem alright. I fail to understand how the remaining two are dx2-y2 and dz2. And how come the quantum numbers -2, -1, 0, 1 and 2 correspond to these?


Because the projections of their angular momentum on a given axis are -2, -1, 0, 1 and 2.

The best thing we can do in order to learn about 4D orbitals is to solve the Schroedinger equation for a hydrogen atom in 4D. No easy job, at least for me, but trying this is on my to-do-list.
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Re: 4D Chemistry

Postby Teragon » Mon Oct 10, 2016 6:27 pm

The first thing we need to do in oder to solve the Schroedinger equation is to calculate the Laplace operator d²/dx² + d²/dy² + d²/dz² + d²/dw² in an appropriate coordinate system. In 4D there are two coordinate system representing the hyperspherical symmetry of the 4D hydrogen atom.

x = r*sin(phi)*sin(theta)*sin(chi)
y = r*cos(phi)*sin(theta)*sin(chi)
z = r*cos(theta)*sin(chi)
w = r*cos(chi)

I'm calling this one the 3-spherical coordinate system.

x = r*sin(theta)*sin(phi)
y = r*sin(theta)*cos(phi)
z = r*cos(theta)*sin(chi)
w = r*cos(theta)*cos(chi)

I'm calling this one the duocirular coordinate system.

Calculations in both will give the same result, but one form may turn out more complicated than the other. I think the duocirular coordinate system is more convenient as it has higher symmetry (the roles of phi and chi are equivalent).

After several sheets of paper I'm still not quite there, but it looks like the resulting differential equation is going to be much more complicated than in 3D, which means it's probably too hard to solve. Let's see...
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Re: 4D Chemistry

Postby quickfur » Tue Oct 11, 2016 6:07 pm

Somebody on this forum has already tried to solve this years ago, and discovered (to all of our chagrin!) that the 4D Schroedinger equation has no minima except at r=0. Which means that atoms as we know them cannot exist in 4D, they'd instantly collapse.

Of course, that is assuming 4D atoms follows an inverse cube law as far as electric charge is concerned; if by fiat we assume inverse square law (which seems a bit hackish since there's no good reason why electric charge shouldn't diminished in proportion to the (hyper)area of a 3-sphere centered on the source), then you might be able to get something interesting.

Note also that technically, in a 2D universe, electric charge ought to obey a reciprocal law (i.e., in proportion to 1/r), and I suspect the 2D Schroedinger equation would have rather unusual solutions in that case. So quantum dots aren't really representative of what would happen in a "real" 2D universe, because, being actually embedded in 3D space, electric charge is "leaking" into the ambient space such that it still obeys an inverse square law, so the solutions to the Schroedinger equation still resemble the 3D case.
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Re: 4D Chemistry

Postby Teragon » Tue Oct 11, 2016 6:53 pm

May I see the derivation?

I see that differently. Our universe only works under very special conditions. It is likely that a 4D universe with exactly the same physical laws doens't work, it might just also have it's own special requirements. Beings in a 4D world thinking about a possible 3D universe, thinking in terms of their own physical laws, might well come to the conclusion that a 3D universe is physically impossible. In our universe the strong force grows with the distance. How weird is that? Coulomb's law is very straight forward in 3D (actually only for large enough distances), but phsyical relationships don't have to be straight forward, often there much more complex. 200 years ago there was also no "good reason" to assume something beyond classical mechanics or something beyond newtonian space time. Realitiy told us otherwise and so we researched until we found a consistent model based only on observations.
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Re: 4D Chemistry

Postby quickfur » Tue Oct 11, 2016 8:02 pm

I don't have it handy right now, you can probably search for "4d atom" or "4d schroedinger's equation" on this forum for older posts, and you should be able to find it. (I wasn't the one who did the derivation, btw.)

At some level, I think we actually agree. Dimensional analogy from 3D to 4D is a convenient clutch for us poor 3D folk to be able to get a grip on how the 4D world might work; and it's always fun to speculate about what might happen if everything about our 3D universe remained the same except for an extra spatial dimension. We get to learn what stays unchanged, and what becomes radically different, and having direct analogy gives us a sense of familiarity so that we're not overwhelmed by the strangeness of it all.

A native 4D universe, however, probably would operate on radically different principles than anything we've imagined. For example, some time ago somebody posted a link to a research article about how signal propagation behaves radically differently in 4D than in 3D. Being an even-numbered dimensionality, 4D behaves similarly to 2D, in that if you have a point source of a wave (the signal), as the wave propagates outwards from the source it will actually send back echoes of itself back to the source, which then interferes with subsequent parts of the signal and combines with it as it bounces back again, causing subsequent parts of the signal to be distorted in complex ways. You can see this in 2D by dropping a pebble into a still pond and seeing how it's not just a single wavefront, but multiple ripples moving back and forth in a complex way behind the initial main ripple. The net effect of this, is that a single peak, say a blip of light from some point source, will be perceived by the receiving end as one big blip followed by "echoes". So in both 2D and 4D, a single flash of light gets perceived as multiple afterflashes. The way the differential equations work out, only in 3D will the receiver be able to easily recover the original signal; in all other odd dimensions, the signal will be recoverable but distorted (IIRC the receiver sees the n'th derivative or integral of the original signal, something like that). In even dimensions, the signal will be quite scrambled, and it's probably difficult, if not impossible, to recover the original signal. This in turn implies that a "native" 4D being would probably not have sight the way we know it, because vision will be inherently blurry due to everything having "afterflashes"! Sound will likewise be "echoey" and "noisy", so aural communication will probably take on strange new forms quite unlike what we're used to.

This, coupled with the Schroedinger equation for the 4D hydrogen atom having no minima except at r=0, means that if matter exists in a 4D universe at all, it would have to be of a radically different nature than our familiar 3D atoms. I wouldn't even know where to begin to speculate what this might look like!

And that's not to mention that there are no stable orbits in 4D (except for the perfect circle, which is so unlikely it probably will never happen -- the slightest deviation causes the planet to either crash into the star or fly off into cold, dark space), which means things like galaxies and solar systems are unlikely to exist in 4D.

But at the end of the day, perhaps none of this is really that important... after all, it's more fun to speculate on what happens in a universe that's basically identical to our own 3D one, except where there's an extra dimension of space to play with, than to try to conceive of something so alien and incomprehensible as a "native" 4D universe that "works" from first principles, when we don't even know what the first principles ought to be!
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Re: 4D Chemistry

Postby The Shadow » Sat Nov 05, 2016 4:42 pm

Teragon wrote:The first thing we need to do in oder to solve the Schroedinger equation is to calculate the Laplace operator d²/dx² + d²/dy² + d²/dz² + d²/dw² in an appropriate coordinate system. In 4D there are two coordinate system representing the hyperspherical symmetry of the 4D hydrogen atom.

x = r*sin(phi)*sin(theta)*sin(chi)
y = r*cos(phi)*sin(theta)*sin(chi)
z = r*cos(theta)*sin(chi)
w = r*cos(chi)

I'm calling this one the 3-spherical coordinate system.


I've seen an article deriving the 4d hydrogen atom using this. (Assuming inverse square propagation, though - as mentioned, inverse cube doesn't have any bound states.)

x = r*sin(theta)*sin(phi)
y = r*sin(theta)*cos(phi)
z = r*cos(theta)*sin(chi)
w = r*cos(theta)*cos(chi)

I'm calling this one the duocirular coordinate system.


They're often called "Hopf coordinates".

Calculations in both will give the same result, but one form may turn out more complicated than the other. I think the duocirular coordinate system is more convenient as it has higher symmetry (the roles of phi and chi are equivalent).


That's exactly what I thought, but when I tried it I got an awful mess. I'm not sure why, you'd think it would work out nicer.
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Re: 4D Chemistry

Postby Teragon » Fri Dec 09, 2016 2:10 pm

The Shadow wrote:I've seen an article deriving the 4d hydrogen atom using this. (Assuming inverse square propagation, though - as mentioned, inverse cube doesn't have any bound states.)


It would be cool if you could find it. The angular part is the more interesting thing anyway. In the mean time I've found an elegant way to generate 1/r-potentials in any number of dimensions greater than two! I'm going to present it in new thread. Of course I don't know how far it is consistent with the rest of 3D- or possible 4D-physics.
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Re: 4D Chemistry

Postby Teragon » Mon Dec 12, 2016 12:33 pm

I know it doesn't really belong here, but just a few words on your comment, quickfur.

quickfur wrote:Being an even-numbered dimensionality, 4D behaves similarly to 2D, in that if you have a point source of a wave (the signal), as the wave propagates outwards from the source it will actually send back echoes of itself back to the source, which then interferes with subsequent parts of the signal and combines with it as it bounces back again, causing subsequent parts of the signal to be distorted in complex ways. You can see this in 2D by dropping a pebble into a still pond and seeing how it's not just a single wavefront, but multiple ripples moving back and forth in a complex way behind the initial main ripple.


Having glanced at the corresponding thread this seems well-founded to me. However I can't understand this analogy argument. Throwing a stone into a pond leads to a vertical elongation of the surface of the water at this location. With gravity as a restoring force it is inevitable that the surface oscillates back and forth, producing a train of circular waves rather than a single pulse. Because of this also in 3D waves are not single peaks, but longer wave trains encompassing several oscillations.

quickfur wrote:The net effect of this, is that a single peak, say a blip of light from some point source, will be perceived by the receiving end as one big blip followed by "echoes". So in both 2D and 4D, a single flash of light gets perceived as multiple afterflashes.


This may be true for sound, but not for light in everyday life. For the same reason that if you stand in a room with mirrored walls and switch off the light it will still get dark immediatly. An impulse of light travels so fast that in the time span of 0.01 s it gets reflected 600,000 times on the walls of a walls of a 5 m diameter room. Even if the mirrors reflect 99.999% of the light and nothing else is in the room, only 0.25% of the initial intensity would be left after 10 ms. In reality light is always emitted from something, so each time the reflected wave hits the source a large fraction of the light will be absorbed.

quickfur wrote:And that's not to mention that there are no stable orbits in 4D (except for the perfect circle, which is so unlikely it probably will never happen -- the slightest deviation causes the planet to either crash into the star or fly off into cold, dark space), which means things like galaxies and solar systems are unlikely to exist in 4D.


Well, that's only true for a 4D universe with otherwise identical physical laws to our own universe. We shouldn't talk about it, as if it was a general property of 4D. A lot of fine-tunig is necessary in 3D to produce a universe where solar systems and stable matter can exist. Adding a forth spacial dimension reshuffles the pack, a 4D universe needs different physics in order to form complex matter.
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Re: 4D Chemistry

Postby quickfur » Mon Dec 12, 2016 5:51 pm

Teragon wrote:I know it doesn't really belong here, but just a few words on your comment, quickfur.

quickfur wrote:Being an even-numbered dimensionality, 4D behaves similarly to 2D, in that if you have a point source of a wave (the signal), as the wave propagates outwards from the source it will actually send back echoes of itself back to the source, which then interferes with subsequent parts of the signal and combines with it as it bounces back again, causing subsequent parts of the signal to be distorted in complex ways. You can see this in 2D by dropping a pebble into a still pond and seeing how it's not just a single wavefront, but multiple ripples moving back and forth in a complex way behind the initial main ripple.


Having glanced at the corresponding thread this seems well-founded to me. However I can't understand this analogy argument. Throwing a stone into a pond leads to a vertical elongation of the surface of the water at this location. With gravity as a restoring force it is inevitable that the surface oscillates back and forth, producing a train of circular waves rather than a single pulse. Because of this also in 3D waves are not single peaks, but longer wave trains encompassing several oscillations.

It's not the fact that it oscillates that's the issue. It's the fact that the relationship between the final waveform and the original signal (presumably some kind of oscillator producing a sine wave, say, at the point of origin) is non-linear. IIRC the paper describes the effect as the net effect of the summation of wavelets that make up the front of the expanding waveform. As the waveform expands, the peaks at its frontier produces wavelets that travel both forwards and backwards; in 3D, however, the backward-travelling wavelets cancel out and so you have a clear sine wave, for example, throughout the entire spherical region from the origin to the frontier. However, in other dimensions, the backward-travelling wavelets do not cancel; thereby producing a residual effect that changes the shape of the tail of the waveform via interference. In even dimensions such as 2D and 4D, this causes every peak of the original sine wave signal to produce backward travelling peaks that significantly alters the overall shape of the waveform, so that the overall shape can no longer be described by a simple sine wave but a complex combination of forward and backward travelling sine waves. In odd dimensions l ike 5D, the backward-travelling wavelets "mostly" cancel out, but not quite, thus leaving their trace in altering the final waveform so that instead of being a faithful representation of the original sine wave signal, it ends up being an n'th derivative of the original signal. Only in 3D is the original signal propagated without being mutilated in some way.

quickfur wrote:The net effect of this, is that a single peak, say a blip of light from some point source, will be perceived by the receiving end as one big blip followed by "echoes". So in both 2D and 4D, a single flash of light gets perceived as multiple afterflashes.


This may be true for sound, but not for light in everyday life. For the same reason that if you stand in a room with mirrored walls and switch off the light it will still get dark immediatly. An impulse of light travels so fast that in the time span of 0.01 s it gets reflected 600,000 times on the walls of a walls of a 5 m diameter room. Even if the mirrors reflect 99.999% of the light and nothing else is in the room, only 0.25% of the initial intensity would be left after 10 ms. In reality light is always emitted from something, so each time the reflected wave hits the source a large fraction of the light will be absorbed.

I think you're misunderstanding what I wrote. :-) The point isn't that the light gets reflected, the point is that the waveform of the light pulse (or steady wavetrain, whichever) takes on a fundamentally different form from the energy source. Thus communicating the original signal becomes difficult or perhaps even impossible, because by the time it gets to the receiving end it has mutated into a very different waveform.

Or perhaps you're taking issue with my interpretation of the equations, which is perfectly understandable. I think at a fundamental level we actually agree. We could save ourselves the trouble if you read the paper yourself and looked at the differential equations and their solutions as presented there; it makes it quite clear that there is something unique about 3D that causes wave propagation to retain the pristine form of the original signal, whereas in all other dimensions some kind of fundamental alteration occurs by the time the signal reaches the receiver, in odd dimensions as a derivative of the original waveform, and in even dimensions a far more complex alteration that likely makes recovering the original signal very difficult, if not impossible.

quickfur wrote:And that's not to mention that there are no stable orbits in 4D (except for the perfect circle, which is so unlikely it probably will never happen -- the slightest deviation causes the planet to either crash into the star or fly off into cold, dark space), which means things like galaxies and solar systems are unlikely to exist in 4D.


Well, that's only true for a 4D universe with otherwise identical physical laws to our own universe. We shouldn't talk about it, as if it was a general property of 4D. A lot of fine-tunig is necessary in 3D to produce a universe where solar systems and stable matter can exist. Adding a forth spacial dimension reshuffles the pack, a 4D universe needs different physics in order to form complex matter.

Of course the context is that we're talking about identical (or at least analogous) physical laws to our own universe here! Otherwise what do planetary orbits even mean?? In a 4D universe with fundamentally different physical laws to our own, it seems meaningless, or at least presumptuous, to even speak of planets, as if they necessarily exist under arbitrary physical laws! For that matter, what does "matter" even mean? How are we so sure that such a thing would even exist in a 4D universe that operates according to completely alien physical laws?
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