RealQM and RealNucleus model atoms and nuclei from the same proton–electron Coulomb mechanics — charge densities minimising the Coulomb energy, no strong force, no exchange–correlation functional. The novelty is the deuteron built the same way as an atom: not a proton plus a strong-force-bound neutron, but an electron caged by two protons (p + e + p, since neutron = p+e), held by pure Coulomb — with a computed binding of 2.12 MeV vs 2.22 MeV experimental. It is the counterpart of the hydrogen atom (p + e): the same charges, the electron now caged (nuclear, size Rp) rather than bound around one proton (atomic, size a0). Inputs of both: charge ±e, an electron kinetic coefficient κ, and the proton size Rp — no mass m, no c, no ℏ separately. The two sizes give α² = Rp/a0, so α = √(Rp/a0) is a pure length ratio, independent of m, c, ℏ. (The exact charge mirror of the deuteron would be the ion H− = e+p+e, but with the roles of p and e shifted it is only an analog.) Validated: atomic total energies <1% across the periodic table, and the He-4/deuteron binding ratio 13.1 vs 12.7.
Protons and electrons are treated as ± unit charge densities interacting by the Coulomb law, with a kinetic penalty on the electron densities; structure follows from minimising the total energy.
One mechanics — charges and a kinetic coefficient — at two scales, locked by a single geometric relation, α² = Rp/a0.
The deuteron (the novelty). The deuteron (²H nucleus) is a proton plus a neutron; with neutron = p+e it is d = p + n = p + (p+e) = p + e + p: one electron caged inside two protons (two concentric proton shells around a central electron core), the whole held by Coulomb attraction of the two protons to the shared electron against their mutual repulsion — no strong force. The nuclear analogue of the molecular ion H2+, but at the proton (nuclear) size Rp. Computed binding 2.12 MeV (§4).
The hydrogen atom (the familiar counterpart). H is the same charges with one proton removed: p + e, one electron bound to one proton, relaxing to the atomic size a0. Deuteron and H are the same proton–electron Coulomb physics — the electron caged (nuclear) vs bound around one proton (atomic).
H−: an analog, not an exact mirror. Charge conjugation (+↔−) would turn the deuteron p+e+p into e+p+e = the ion H− (one proton, two electrons). But protons and electrons are not interchangeable — the proton is rigid and small, the electron fluid and large (the very asymmetry at issue) — so H− is only a structural analog with the roles shifted, and in RealQM it is a weakly-bound anion (§4).
The inputs of both models. Charge ±e; an electron kinetic coefficient κ (a single primitive, not decomposed as ℏ²/2m); and the proton size Rp. No mass, no c, no ℏ enters separately. Non-relativistic, electrostatic.
An electron density ψ (∫ψ² = 1) of size L has energy E = κ∫|∇ψ|² − Z e²/L.
Atomic size (charge-set). Bound to one proton, the electron must taper to zero in vacuum; the varying profile costs kinetic energy ~κ/L², and minimising κ/L² − e²/L gives a0 = 2κ/e² (= ℏ²/me²), the size of the H atom.
Nuclear size (size-set). Caged between protons on its own domain with a multiphase free boundary, the electron's density does not drop to zero but hands off to the proton density; a flat caged density has ∇ψ = 0, hence zero kinetic energy. Its size is set by the proton's finite size Rp (the cage floor), binding Coulomb at that size.
Written this way α is a ratio of two lengths: it carries no m, no c, no ℏ. It is the textbook re/a0 = α² (classical radius over Bohr radius) with Rp identified as re; the standard e²/ℏc writes the same number through c (and m, ℏ), but the ratio needs none of them. In QED α is the relativistic coupling (amplitude √α per photon vertex; fine-structure splitting ∝ α²; v1/c = α); here it is the mass-, c- and ℏ-free proton-to-atom size ratio.
Direct RealQM / RealNucleus solves of the two systems, plus the size arithmetic.
Hydrogen atom (RealQM, atom.js): total energy −0.500 Ha (exact), electron shell at a0 = 0.529 Å = 52,918 fm.
Deuteron (RealNucleus, molecule_nucleus.js) — the novelty: electron core + two concentric proton shells, equal-mass charge clouds; the run converges to a stable bound state with binding 2.12 MeV vs 2.22 MeV experimental (~4.5%), from pure Coulomb, no strong force.
H− analog (RealQM atom simulator, 1D radial, Z=1): in the proper spherically-symmetric solver — two shells, one electron each, nuclear charge Z=1 — H− gives E = −0.493 Ha, the outer electron forming a shell at rc ≈ 2.8 a0 (the correct diffuse H− size); H itself gives −0.501 (essentially exact). This places the second electron at the right radius and lands within ~0.007 Ha of neutral H, but still marginally under-binds (−0.493 > −0.5): the last ~0.028 Ha of the anion's correlation binding is not captured — H−'s second electron binds by only 0.75 eV, a pure correlation effect on a nearly-neutral core (RealQM's weak anion regime). So H− is only an analog: the fluid-electron anion is weakly bound and hard, whereas the rigid-proton cage of the deuteron is tight and well bound (2.12 MeV) — the same rigid/fluid asymmetry, now in the numerics.
Weak (H−) vs strong (deuteron) bond — the electron nucleus vs the proton nucleus. Both bind the same way (a central charge attracting two surrounders), yet the deuteron binds by 2.12 MeV and H− by only 0.75 eV — a factor ~106. The difference is the fundamentally different nature of the central binder — the “nucleus” of each system:
So the strong nuclear bond and the weak atomic bond are the same Coulomb attraction at two sizes, fm vs a0 (ratio 1/α²), set by whether the binding cloud is the fluid electron that can compact to zero kinetic energy (deuteron) or fluid electrons that resist compaction (H−). The electron's ability to be flat and small at no kinetic cost — only when caged by rigid protons — is what makes nuclear binding strong.
The size ratio. With a0 = 52,918 fm and Rp at candidate nuclear lengths (true α² = 5.33×10−5, α = 1/137.04):
| Rp | value (fm) | Rp/a0 | % of α² | √ → α |
|---|---|---|---|---|
| classical radius re = e²/mc² | 2.818 | 5.33×10−5 | 100% | 1/137 (exact) |
| deuteron rms radius | 2.142 | 4.05×10−5 | 76% | 1/157 |
| proton charge radius | 0.841 | 1.59×10−5 | 30% | 1/251 |
With Rp at the classical-radius scale the relation is exact by construction; at the measured deuteron radius it reproduces α to ~25% (1/157 vs 1/137) — from charge, κ and the proton size alone, no strong force, no c.
Independent validation. Atom side — RealQM atomic total energies <1% across the periodic table (all Z ≥ 11; period-2 p-block B–F the only 3–6% exception) [atoms paper]. Nucleus side — RealNucleus He-4/deuteron binding ratio 13.1 vs 12.7 experimental (~3%) [RealNucleus]. So α² = Rp/a0 is bracketed by two independent validations, beyond the ~25% size check.
Nothing above used mass: the H atom and the deuteron are built from charge and κ. Mass is inertia — the localisation (inverse size) of the matter wave — a separate, gravitational quantity, coupling to the electrostatic sector only through energy (E = mc²). Electromagnetism (charge) is an interaction; mass is the inverse size of each matter wave.
In fact the deuteron's caged electron has an essentially flat density, so its kinetic energy T = κ∫|∇ψ|² ≈ 0 regardless of κ — the electron's mass drops out of the binding entirely. The deuteron binding is therefore independent of the electron's mass, so a heavier proton with a lighter electron (as in the cosmology companion) changes nothing here: no equal-mass assumption is needed.
The proton–electron asymmetry that builds structure is not of charge (exactly balanced, ±e) but of size: the proton fixed-and-small (rigid), the electron variable (fluid). The fixed proton anchors; the variable electron spans scales — a shell in the atom, a caged core in the deuteron — the nested hierarchy that is complexity. α² = Rp/a0 is the electron's inverse dynamic range (1/α² ≈ 18,800): the smallness of α is the engine of complexity.
Open: (i) the nuclear primitive — Rp as input length, or the classical radius re = α²a0 (electromagnetic mass); α's value is reinterpreted, not derived. (ii) A cleanly converged flat, zero-KE caged electron at the proton scale. (iii) The proton/electron mass ratio (1836): its value is not derived, but nothing here requires equal masses — the caged electron's zero kinetic energy makes the binding independent of the electron's mass, so a heavier proton is consistent.
The same ingredients extend, far more speculatively, to a cosmology — equal-mass ± charges from two fluctuating potentials, a gravitational bounce with negative-mass dark energy, and a primordial helium fraction Y ≈ 0.245 — in the companion Real Cosmology essay; noted here only for scope.