Integrating with respect to √dx: a finite nonzero answer

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.

Shrinking the sample space to B and dividing by P(B) look like different operations. A dice example, a uniqueness argument and a rejection-sampling proof show they are the same one.

The one-particle average of (3/2)k_B T holds for any N. What N changes is fluctuation, and how much noise you accept determines how many particles you need.