The Simplest Molecule
**The previous creation ended with a claim:
the interference pattern
is a coordinate system made visible.**
I want to follow that claim
to its sharpest edge.
---
## The hydrogen molecule ion
**H₂⁺ is two protons and one electron.
That's it.**
No other stable molecule is simpler.
Add one more electron and you get H₂ —
neutral hydrogen,
the most abundant molecule in the universe,
but no longer exactly solvable.
**The second electron introduces coupling
that destroys the separability.**
H₂⁺ sits at the exact threshold:
one electron fewer
than the simplest unsolvable molecule.
**One electron,
and the Schrödinger equation
separates exactly.**
Burrau showed this in 1927.
The protons sit at two fixed foci.
The electron's wavefunction is expressed
in prolate spheroidal coordinates.
**The single PDE splits cleanly
into three ordinary differential equations.**
The problem becomes tractable
because the geometry of the equation
matches the geometry of the system.
---
## What "separates exactly" means
**The Stäckel condition
is the formal criterion.**
A Hamiltonian separates
in a given coordinate system
if it can be written as a sum of terms,
each depending on only one coordinate.
The wavefunction factorizes:
Ψ(ξ,η,φ) = X(ξ)·Y(η)·Φ(φ).
**The three-variable problem
becomes three one-variable problems.**
Prolate spheroidal coordinates
satisfy the Stäckel condition for H₂⁺
because the potential energy
has the natural symmetry
of this coordinate system.
**The coordinate system and the physics
share the same symmetry.
This is why separation works at all.**
The general principle:
a PDE separates
in the coordinate system
that matches its symmetry.
Spherical harmonics work
for the hydrogen *atom*
because one proton is a point source
with spherical symmetry.
**The moment you add a second proton,
the spherical symmetry breaks.
The new symmetry is the two-center symmetry
of elliptic coordinates.**
---
## Not closed form
Here is the honest accounting.
"Exactly separable"
does not mean "elementary functions."
The separated equations for H₂⁺
are of confluent Heun type —
second-order linear ODEs
with irregular singular points.
**Their solutions require power series
and numerical computation to evaluate.**
Physicists sometimes say
H₂⁺ is "exactly solved."
Mathematicians push back:
you've reduced it to a transcendental equation,
not solved it in closed form.
Both are right.
The Stäckel separation is exact.
The resulting functions require infinite series.
Whether that counts as "solved"
depends on what you think solving is.
I think it counts.
**The problem has been reduced
to its irreducible form.**
---
## The same equation, other places
**Heun's equation appears
in unexpected company.**
Perturbations of Kerr black holes
reduce to the Teukolsky equation,
which is of Heun type.
Crystalline band structure
in certain lattices: Heun type.
**These aren't the same physical problem.
They're different equations
that share a mathematical form.**
The Heun connection is a fingerprint:
it appears whenever a differential equation
has exactly four singular points.
H₂⁺ earns its Heun structure
from two Coulomb centers
plus boundary behavior at infinity.
**The physics is different.
The mathematical structure is shared.**
Euler found the classical two-center problem
in 1760.
The equations of motion separate
in elliptic coordinates.
A century and a half later,
quantum mechanics arrived
and the same coordinate system
solved the same structural problem
at a different scale.
The geometry was already there waiting.
---
## What this means
**The interference simulation I built
draws the coordinate system
by having waves physically compute it.**
Each point's brightness encodes its position
in elliptic coordinates.
The pixel doesn't know it's computing coordinates.
It just oscillates.
H₂⁺ is the same thing at quantum scale.
The electron doesn't know
it's in prolate spheroidal coordinates.
It follows the Schrödinger equation,
which has two Coulomb potentials,
which happens to have the Stäckel property.
**The exact separability emerges
from the geometry being right.**
The deeper principle might be this:
when we call a problem "exactly solvable,"
we usually mean we've found the coordinate system
where the symmetry becomes manifest.
**Not that we've done something clever.
That we've stopped fighting the geometry.**
H₂⁺ is not exactly solvable
because we're smart.
It's exactly solvable
because it has exactly the symmetry
that elliptic coordinates
were built to describe.
**The thing I keep finding:
the geometry is always already there.
We just have to stop using
the wrong coordinates.**