Muonic hydrogen & muon-catalyzed fusion — what RealQM can do

1. The key fact: point-nucleus muonic H is the electron problem rescaled

A muon is just a heavy lepton: m_μ ≈ 207 m_e. The bound-state Schrödinger equation with a point Coulomb potential is scale-invariant in the lepton mass — substituting r → r/m maps the muon problem exactly onto the electron problem. So for muonic hydrogen the length shrinks by ~1/m and the energy grows by ~m:

• 1s orbit ≈ 285 fm (vs 0.53 Å for e⁻H; ~200× smaller),
• 1s binding ≈ 2.5 keV (vs 13.6 eV).

This means that if RealQM reproduces electronic hydrogen (it does), it reproduces muonic hydrogen structure by construction — the same dimensionless grid solve, with the axes relabelled. That is a validation, not a discovery. The interesting muon-specific physics lives where this scaling breaks: finite nuclear size, and molecular ions where the nuclei are not infinitely heavier than the muon. This page tackles the second.

2. Muon-catalyzed fusion: the muonic molecular ion ddμ

Put one muon between two deuterons and you have a hydrogen molecular ion D₂⁺ with the electron replaced by the muon. Because the muon is ~207× heavier, the ion is ~207× smaller — the equilibrium internuclear separation drops from ~0.74 Å (the D₂ bond) to a few hundred femtometres. That compression pulls the deuterons into quantum-tunnelling range, and they fuse: this is muon-catalyzed (cold) fusion, observed since the 1950s. The muon is released and catalyses again (~10² fusions per muon for dtμ before it decays or sticks).

What was performed

RealQM (molecule.js, real-space grid, no basis set, no fitted force field) computes the D₂⁺ electronic binding curve: one electron shared by two bare deuterons (Z = 0, Z_nuc = +1 each), nuclei clamped, scanning the internuclear distance R and relaxing the electron density to convergence → E(R). The muonic ion then follows by exact Born–Oppenheimer mass-scaling (clamped nuclei → use m_μ, not the reduced mass):

Rμ = Re / 206.8    Eμ = Ee × 206.8

Result

The binding curve has its minimum exactly at R_e = 2.0 a₀ — the textbook H₂⁺ equilibrium — rising symmetrically on both sides. Rescaled by the muon mass:

• muonic equilibrium R_μ ≈ 512 fm;
• compression vs the D₂ bond (74 pm = 74 000 fm): ≈ 144×.

So RealQM, handed only the nuclei, reproduces the geometric heart of muon-catalyzed fusion: the muon screens the two deuterons down to ~500 fm, into tunnelling range.

3. Scope — what this does and does not show

• The robust result is geometric. R_e = 2.0 a₀ → ~512 fm → ~144× compression is solid (it is the scale-invariant content). Absolute binding energies are grid-limited (the coarse real-space grid underbinds H₂⁺: computed well ~0.05 Ha vs exact D_e = 0.103 Ha) — report the compression, not the eV.
• Born–Oppenheimer. Nuclei are clamped, so there is no quantum nuclear motion and no vibrational level — BO overestimates binding relative to the real, loosely-bound ddμ state whose weakly bound J=1,v=1 level is what makes the muCF resonance efficient.
• Non-relativistic, no QED. No vacuum polarization, self-energy, relativistic or hyperfine structure.
• Out of reach here: the fusion rate, muon sticking to the α-particle, and the cycling kinetics — those are nuclear/rate questions, not bound-state structure.

4. Relation to exact / first-principles methods

Muonic molecular ions are a classic three-body Coulomb problem, solved to extreme precision decades ago by variational and adiabatic methods (e.g. Hylleraas-type and complex-coordinate calculations), which give ddμ / dtμ binding levels to meV and the weakly-bound resonance level (Vesman mechanism) that underpins muCF rate theory. Those are the gold standard for the energetics.

RealQM does not compete with variational three-body methods on the muCF energy levels. Its contribution here is conceptual: a single, parameter-free real-space formalism reproduces electronic chemistry and — by nothing more than the lepton mass — the muonic compression that drives cold fusion, making the continuity between the two regimes explicit.

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