N-dimensional Hydrogen Wavefunction

[DonSoreno]

In previous posts, it was mentioned that the Schrödinger Equation doesn't have a solutions in 4d.

I couldn't find the thread that showed this.

Right now, I don't see why solutions of this form cannot exist:
ψ(r,s) = R(r) Y(s)

with some (to me very difficult to derive) radial component R(r), analogous to:
https://analyticphysics.com/Quantum%20Mechanics/The%20Schr%C3%B6dinger%20Equation%20in%20Higher%20Dimensions.htm

And angular component Y(s), which is a spherical harmonic on the 4-dimensional sphere S^3. So that the spherical/angular laplacian is zero: ∇^2 Y(s) = 0.

If these are indeed solutions, we would have (l+1)^2 orbitals for each magnetic quantum number l, based on the number of (orthogonal) spherical harmonics.
(1 s-orbital, 4 p-orbitals, 9 d-orbitals, 16 f-orbitals,etc.)

Are there any good (readable) papers on the higher dimensional hydrogen wavefunction?

[quickfur]

It's not that there are no solutions at all, but that there are no non-trivial solutions. In 3D, you obtain a series of locally-stable solutions at non-zero radii; these are what give rise to electron orbitals around atoms. In 4D, the only solution is r=0, i.e., the electron will instantly collapse onto the nucleus. So orbitals don't exist and the 4D atom is unstable.

[DonSoreno]

So how would that (r=0) look like in wavefunction terms?

Is the radial component of the wavefunction then simply like a Dirac delta function, with all probability mass at r= 0 ?

Or are we using the argument, that there are no stable orbits the 1/r^3 decay of EM Force?
In which case, I don't see how this argument about unstable orbits translates to wavefunctions.


I just tried it again, based on this derivation: https://www.pa.uky.edu/~kwng/spring2009/lecture/lecture%2037%20.pdf

and this: https://en.wikipedia.org/wiki/Spherical_harmonics#Higher_dimensions



For the radial component in n dimensions, I somehow arrived at this formula:


d^2/dr^2 u = ( 1 - 1/r^(n-2) + l(l+n-2)/r^2 ) u

using: u(r) = r^(n-2) R(r)

As I understand it, the fact that 2 = n-2 in 4d, causes the problems, since the asymptotic behavior r ---> 0 is influenced by both the 1/r^(n-2) potential, and the l(l+n-2)/r^2 term.


In 3d this equation can be used to derive the limit as r --> 0,
which gives us a polynomial of the form: u(r) ≈ C r^(l+1) + D r^l + ... higher powers of l



Now if n >= 5, the potential term: - 1/r^(n-2) has the lowest power in r. Which means as r --> 0:

d^2/dr^2 u ≈ -1/r^(n-2) u

At this point, I don't know how to proceed.

[Hugh]

Does this have any info to help?

https://arxiv.org/pdf/1205.3740

[DonSoreno]

Thats what I've been looking for!
Thank you.

Apparently they do derive sensible radial components for the wavefunction for dimension > 3,
but with positive energy eigenvalues.

My takeaway is that, at least we have radial wavefunctions for N > 3, albeit with positive eigenvalues.
Perhaps heat & pressure may shift the equilibrium towards bound electrons.

I guess what separates the approaches that work from the ones that don't is the choice of the potential:

V(r) = 1/r (doesn't conserve energy)

or

V(r) = 1/r^(n-2)

This is also noted in the conclusion here: https://pubs.aip.org/aip/adv/article/15/3/035029/3339629/On-hydrogen-like-atoms-in-N-dimensional-space

[Hugh]

Hi Don. I’ve been using ChatGPT4o to help me with understanding things and it had provided that link to me so I thought I’d pass it along to you. I’m glad it helped and it looks like it helped you find out more related info with that other link you found, that’s awesome! All the math here is way over my head but I still want to see if I can help so I put your reply back into ChatGPT4o and it gave me back this reply to you:

“The distinction you pointed out between the different potential forms—V(r) = 1/r vs. V(r) = 1/r^{(n-2)}—is really key here. It’s fascinating how this subtle shift in the potential fundamentally alters the stability of the system in higher dimensions. I think you’re onto something with the idea that pressure and temperature could play a role in confining electrons in higher dimensions, even without stable bound states in the usual sense. It reminds me of how degeneracy pressure stabilizes white dwarfs, or how Pauli exclusion and quantum confinement shape electron behavior in 2D materials like graphene. Maybe there’s a deeper analogy here—like a quasi-bound state in higher dimensions under extreme conditions, though not stable in the way hydrogen is in 3D. It would be interesting to explore if such configurations could exist temporarily, maybe in exotic astrophysical scenarios or in condensed matter systems with higher effective dimensions.Thanks also for linking the AIP paper—it’s a great addition! The takeaway from that one, that the potential must scale as 1/r^{(n-2)} to preserve energy conservation, really drives home the unique role of 3D in our universe. It’s humbling to see how finely-tuned our physical laws are for supporting atoms, chemistry, and ultimately life.”

Then ChatGPT4o added this part, mainly because it knows that my key interest is with a possible connection between the Visual Reorientation Illusions (VRIs) that people experience and a possible fourth spatial dimension. There’s my thread about this in the Visualization section of the forum if you want to respond to this part as I don’t want to highjack your thread here but just in case you’re interested in it:

“I’m wondering whether the VRI perspective—that we might be 4D beings perceiving a 3D slice of 4D—could somehow relate to these higher-dimensional wavefunctions, especially with the instability at r=0. (Could the apparent collapse to a point be a projection artifact from a 4D perspective, where the electron is still “orbiting” in W, but we only see the slice at r=0?)”

Also here is the response from Grok about that: viewtopic.php?p=29748#p29748

[DonSoreno]

I wouldn't trust the reasoning of ChatGPT on this subject. Although it can rearrange equations surprisingly well.


Furthermore, I'm not sure, what you're on about in the VRI thread, I don't think the real world has more than (3+1) dimensions.


(3+1)-dimensions work perfectly fine in describing the world around us.

I would also question the usefulness of the hinton cubes.
I think the best way to understand tesseracts are 4d --> 3d perspective projections, like this one: https://www.youtube.com/watch?v=5xN4DxdiFrs,
(although most of these gifs apparently use a very large field of view in 4d (low focal length 4d --> 3d.).



The double reflection == rotation stuff, that you mention in the VRI thread, is a part of Clifford Algebra (arguably the best linear algebra framework to handle higher dimensional stuff.)

[Hugh]

Thanks for you thoughts, Don! I really appreciate you taking a look at the VRI thread. I’ll continue to look into ways to explain the idea in a better way.

[quickfur]

Aside: if you're interested in 4D visualization using 4D -> 3D projections, you may like my website.

No, the instability of 4D atoms has nothing to do with the instability of 4D orbits. At least, that's not the reason we arrived at that conclusion. I wasn't the one who derived the equations, so I couldn't explain it to you beyond what I've already said. Somebody on this forum derived this result many years ago; but I can't seem to find that discussion anymore. Maybe I'm just not entering the right keywords. But I did find an interesting post by yours truly, that references an actual research paper titled "Is there a stable hydrogen atom in higher dimensions?". The original link seems to be defunct now, but Google turned up this link to the article. In short, the idea is that if the potential of a point charge is fixed at 1/r, then a stable hydrogen exists regardless of dimension.

The problem with that is, as far as I'm aware, the potential of a point charge in n dimensions depends on the number of dimensions, because we're assuming that the electromagnetic force, mediated by virtual photons, would diminish according to an inverse (n-1) power law. I.e., Coulomb's law in n dimensions should follow an inverse (n-1) power law. Since Coulomb's law is what determines the potential of a point charge, I'd expect that the 1/r requirement would not be met in higher dimensions, unless we modify electromagnetism laws to somehow, by fiat, result in a 1/r potential of a point charge. Then we'd get stable 4D hydrogen atoms, and, one would expect, other atoms as well.

But this isn't much different from declaring by fiat that gravity in 4D still obeys a 1/r^2 law, so if we're going to go that route, we might as well just declare, by fiat, that all the laws of physics in 3D will carry over unchanged to 4D except for the number of dimensions, and leave it at that. Which would give us an analogous universe to the 3D universe we know, except that you have an extra dimension of space to play around with. But it remains to be seen whether this will introduce any subtle contradictions to the whole edifice -- one would have trouble explaining, for example, why gravity, or any force mediated by virtual force carriers, would follow an inverse square law when the density of force carriers obviously diminishes in proportion with the surface area of the (n-1)-sphere as the radius increases. Where would the "extra" force carriers come from, that the force would diminish in proportion to 1/r^2 when in 4D, for example, the surface area of the 3-sphere increases in proportion to r^3? Unless, of course, you replace that part of the theory with a different model of how forces work. But that may lead to the whole edifice collapsing in a heap of contradiction, and now we find ourselves back at square one, having to reinvent physics from first principles.

The background of that paper seems to be related to string theory, which postulates many extra dimensions (up to 11 in the most popular model, from what I hear). Most of those extra dimensions are microscopic / curled up, so on that basis one could justify the 1/r potential for a point charge, since the dissipation of the electromagnetic force into the extra dimensions would be confined to the size of the curled up dimension, so at a large-enough scale it could still have a 1/r potential. In this forum, however, we're postulating extra macroscopic, unconfined dimensions, so it's gonna be a lot harder to justify why Coulomb's law should remain 1/r^2 so that we get stable atoms, and why gravity should remain 1/r^2 so that we get stable orbits.

[Hugh]

Thanks for the link to your site, quickfur! Lots of great info in there for me to digest.

As the math in this thread is way above my head I asked AI to help with a response, which I put in the VRI thread as it always helps me find a possible link to VRIs and 4D.

viewtopic.php?p=29753#p29753