This is very interesting stuff! Questions like these could have big significance concerning other possible universes out there!
So the equation of the electric potential is φ(r) = –[q/(4πr)] cos(ωmr). Provided the cos-factor is the same for any number of dimensions, we can say that, if ωm (whatever this constant means - I've found no explanation in the text, but it should be related to the photon mass) is very small, the oscillation period goes towards infinity and the potential will be the same as with massless photons.
If the "wavelength" of the potential is in the range of everyday objects, particles would feel periodic attraction and repulsion in an electric field. For example an electron approaching a charged object of any polarity would feel alternating acceleration and deceleration in increasing magnitude, until it finally gets repelled (provided it doesn't lose energy in the right instance). The larger the kinetic energy of the electron, the closer it gets until it gets repelled. The real difference between a positive and a negative electric field only occurs when the electron has enough energy to get to the region within one wavelength of the E-field. Then opposite charges are attractive and a alike charges are repulsive. The stronger the field, the more energy the electrons needs not to bounce off. On the atomic scale, there is still no difference compared to classical electrodynamics.
If the wavelength is in the range of the atomic radii, I see two problems for the stability of (classical) atoms in 4 or more dimensions. First, the cos makes the decrease of the potential faster, faster even than any inverse power law could do. I have no proof, but if I remember correctly, it is known that any potential decaying faster than 1/r doesn't allow for bound states. Second, if the energy of an electron is higher than the second potential minimum, the electron will tunnel into it and the atom will decay. So this sounds bad, right? Wait, there is a new possibility!
The potential minima following the first one have a very different structure! They're looking more like a cos. We can expand the potential around the nth minimum into
φn(rn) = -q/(8π²nωm) + qrn/(16π³n²ωm²) + qrn²((ωm4)/(16π³nωm³)
That means, close to the minima the potential can be approximated by this formula. rn is the radial distance relative to the nth minimum. The first term is just a constant. There are correction terms that I've left apart, but in a good approximation, the constant of the nth minimum is proportional to 1/nd-3, where d is the number of dimensions. The third term is proportional to x², thus equivalent to hook's law implying stable orbits. I'm not sure if the second, linear term might cause problems, but I would quess, no. As you can see, while the quadratic term decreases with 1/n, the linear term decreases with 1/n², so for larger n, the quadratic term dominates.
This means that some form of atoms might exist in any number of dimensions assuming a very different form than our familiar atoms.
- Electrons would not encompass the whole space around nuclei, but move in more defined orbits with nothing in between.
- Their stability and the possible wave functions would critically depend on ωm (in relation to Plank's constant)
- For atoms with only one electron, the orbits would exhibit relative energy spacings proportional to 1/n. Every additional electron has the effect of shielding the positive charge of the nucleus, accelerating the decrease of the series. If the electron distribution is not spherically symmetric, like it is often the case in matter, this might have unforeseeable consequences, including a destabilization of the involved atoms.
- Within an orbit, there would be a set of different possible energies with equal spacing (1/2 + m).
- For each set of (n,m), there would be different wavefunctions defined by the spherical harmonics in d dimensions, yielding (d-1) additional angular quantum numbers. Their number would not increase with n in the same way it does in 3D, but likely be much larger and depend on ωm.
- Excitations to orbitals with higher n induced by light are much more unlikely then excitations between different orbitals in common atoms. The reason is that due to the missing overlap of the wave functions, no direct transition is possible. Instead, electrons will be excited to metastable states from where they can tunnel into the next higher orbit. All other transitions between shells are even much more unlikely!
- Conversely excitated states of atoms would be much more stable, because it takes an additional tunneling process in order for the atom to lose energy. If an electron on a deep orbits is kicked out, the result is a cascade of tunneling processes, each one accompanied by the emission of a photon.
- Bound states could exist both between opposite and alike charges, e.g. between protons and positrons or between two electrons.
- In 3 (and 2) dimensions, in addition to the shell electrons, some electrons could exist in the central potential minimum. Those might behave very similar to the electrons in common atoms.
- Chemistry would look very different than in 3D. My gut feeling is that it will be less complex and bonds will be less stable. However I might be totally wrong with that suspicion. What is for sure is, that the math will be much more complicated.
So far my deliberations. I might be wrong in one point or another.
What is deep in our world is superficial in higher dimensions.