Probability Current of a Plane Wave, Explained

We check a Master Theorem construction for f^s(x) against affine maps by hand, then see where convergence and uniqueness give out.

Replacing dx with √dx in a Riemann sum multiplies the ordinary integral by √n. A finite answer therefore needs a zero integral and a precisely tuned singularity.

An expected dipole moment that oscillates at the transition frequency, fed into the Larmor formula. The result is close to the quantum emission rate, but a single factor shows where the classical argument breaks down.