AI vs Quantum Mechanics

The Position After a Hundred Years

Schrödinger wrote his equation in 1926. A century later it is still believed exact for atoms and molecules, and it is still unsolvable — not difficult, unsolvable. The wave function for N electrons lives on 3N variables, so the cost of holding it grows exponentially. Helium can be done to many digits. Benzene has forty-two electrons and a wave function on a 126-dimensional space, and no computer that will ever be built can hold it.

What a hundred years has delivered, then, is not solutions but a ladder of substitutes. Coupled cluster is reckoned the gold standard and scales as the seventh power of system size, which means tens of atoms. Density functional theory reaches thousands of atoms and carries essentially all of computational materials science — but before any number comes out you must name an exchange–correlation functional, and there are several hundred, most containing parameters fitted to measured data. The theorem says the exact functional exists. It does not say what it is.

So: the accurate end does not scale, and the scalable end is fitted.Add basis sets, dispersion corrections and pseudopotentials, each calibrated against measurement, and you have the patchwork — a century of careful, honest, indispensable repair work, every piece of it needed because the thing underneath cannot be computed.

This is also why the theory is hard to test. A claim is tested only if it could fail. When a calculation disagrees with experiment there is a functional to change, a basis to enlarge, a correction to add; and when it agrees, agreement was available by construction, because the ingredients were chosen against measurements of the same kind. That is not bad faith. It is the structural position of a theory whose exact content is out of reach.

 First Question: Can AI Clean Up the Patchwork?

Partly — and it is already happening. Neural exchange–correlation functionals exist and fix known failure cases that defeated hand-built ones. Neural-network wave functions go further and skip functionals altogether, using a network as the trial function in a variational calculation: no functional named, a genuine upper bound on the energy, accuracy competitive with the best conventional methods on small systems. Anyone who says machine learning has added nothing here is not looking.

But notice what this does and does not do. It compresses the patchwork. One learned object replaces many hand-crafted ones. What it cannot do is remove the reason the patchwork was needed, because the 3N-dimensional problem is still there and any method that scales is still an ansatz or a functional. A learned functional is fitted — fitted better, on more data, with more care, but fitted. A hundred small fits become one large fit.

And the trade has a cost that is rarely counted. The patchwork is at least legible. Each approximation has a name, a published derivation, a known failure mode, and a user who can say why it was chosen. A neural functional has none of that. You have exchanged a hundred inspectable approximations for one you cannot inspect, and the epistemic position is unchanged: there is still an ingredient standing between the theory and the number, so the theory still cannot fail cleanly.

So the answer to the first question is: AI can tidy the patchwork. It cannot abolish it. Cleaning up is compression, not derivation.

A Different Bet

RealQM wagers differently. Instead of one function on 3N dimensions, it uses N charge densities on ordinary three-dimensional space, each carrying one unit of charge, occupying regions that do not overlap, with the boundaries between them free and located by minimising the energy. The energy is a kinetic localisation term plus the Coulomb energy of the charge configuration, each pair counted once. No antisymmetry, no exchange, no correlation functional. The exclusion principle is not an algebraic condition on a wave function but the geometric statement that two electrons do not occupy the same place.

Whether that is the right physics is precisely the open question. What matters here is that it is a computable question. Cost grows with the number of electrons and the fineness of the grid, not exponentially. Given the nuclear charges and their arrangement, the energy is fully determined: there is nothing left to choose, and so nothing left to adjust when the answer comes out wrong.

Second Question: Can AI Realise the Model?

Here the answer is yes, and for a dull reason that turns out to be the important one.

RealQM's bottleneck has never been ideas. It has been the labour of computing consequences and checking them — building solvers, imposing the right conditions at the free boundaries, running the grid ladders, comparing against exact values and against measurement. That labour is exactly what has become cheap. Constructions that would each have taken weeks now take hours, which means claims can be confirmed and, more usefully, killed at a rate that was not previously available.

The condition is that every claim be pinned to something outside the calculation that produced it: an exact value, a measurement, a halved grid, a number computed by different code. Fluent output is not evidence. But those anchors are available here, and that is not an accident — they are available because the model is computable and has nothing left to choose.

Why the Two Questions Have Different Answers

AI accelerates whatever it is applied to. Applied to a theory whose usable versions are all parameterised, it accelerates parameterisation: better functionals, better surrogates, better fits. That is valuable engineering and it tells you nothing about whether the theory is right, because nothing in it was ever at risk.

Applied to a theory that can be computed outright, it produces verdicts — and some of them will be unfavourable. That is the point, not a drawback.

So AI does not make RealQM correct. It makes RealQM decidable in practice. And a theory that can be decided is worth more than one that cannot, whichever way the decision goes.

Where It Stands

Plainly: the geometry comes out well and the energies not so well. Structures, bond angles and the direction of forces are reliable, which is the regime where atoms merely rearrange. Absolute energies carry errors of a few per cent at best, and differences taken across different compositions are not trustworthy. The equilibrium coefficients of continuum mechanics follow directly from computed energies with no averaging of any kind — a real result, and the thing the model does that the standard route cannot do without an ensemble. Dissipation does not follow at all, and is the open problem.

None of that is settled. All of it is now testable at a rate that makes settling it a realistic prospect rather than a programme for the next generation. After a hundred years of necessary repair work, that is the change worth noticing.