Quantum drift · ⛓️ · slide the atoms, pour the electrons
checking…
THE CRYSTAL slide the atoms together · each level fans into a band · pour electrons · read the lamp
↔ drag the atoms together
—
pour electrons to find out
electrons 0 / 0
Self-test — the proof
A periodic crystal (Kronig–Penney). Each line is a falsifiable claim — the dispersion checked against an independent transfer-matrix and a from-scratch ring eigensolve, the metal/insulator verdict checked as a pure occupancy fact.
One atom holds sharp energy levels. Line up a row of them and slide them together:
each level can't stay sharp — it fans open into a band, and dark forbidden gaps tear open
between. Pour electrons in. Whether the row is a metal or an insulator comes down to one
thing you can watch: does the top band sit half-full, or top off right under a gap.
The goal
Make this crystal a METAL, then an INSULATOR, without changing the electron count — only the lattice.
Slide the atoms together
Pour the electrons
allowed bands—
states / band (8 atoms × spin)2N = 16
band 1 width (it fans)—
electrons poured0
top band fill—
gap above it—
edges: f(E)=±1 vs ½·tr M(E)—
For the rigor-minded
|f(E)| ≤ 1 → allowed band (green) · |f| > 1 → forbidden gap (amber). The band edges drawn on the crystal are exactly where this curve touches ±1.
A band holds 2N states (a ring of N=8 atoms, ×spin). Fill a band exactly to the top and the
next electron must clear a gap — no states to move into, so it can't conduct:
INSULATOR. Stop a band half-full and electrons slide freely: METAL.
Crank P high (atoms held apart) and every band shrinks to a hairline at the isolated level En=n²π²/2a²;
drop it toward 0 and the gaps slam shut — then it's always a metal, the honest
negative control. The fan you watch is the real computed band, proven two ways.