QUANTA

E = hf − φ — below threshold, a floodlight ejects nothing; above it, a glimmer.

Only the rungs exist — and the jumps between them are Balmer’s lines, to the nanometer.

Δλ = λ_C(1−cosθ) — and the recoil electron’s E²−p² is still m², to the bit.

Two 511 keV photons, exactly back-to-back — the 180° derived from p-conservation, residual 0.

λ = h/p: an electron at 100 eV waves at 1.22637 Å — atom-sized, and the microscope is born.

One exact stationary wave, matched twice: density may decay inside the barrier while probability current crosses and R + T stays one.