quantum mechanics - Relation between radio waves and photons generated by a classical current

Several questions have been posted on Physics SE regarding the relationship between photons and electromagnetic waves, and several good answers have been given. Some of those questions are listed below, but I didn't find any that requested a mathematically explicit analysis of what happens — in terms of photons — when an oscillating current generates an electromagnetic wave with a macroscopic wavelength, such as a radio wave.

I'm attempting to fill that gap by posting this question-and-answer.

I haven't found an equally explicit / equally narrated analysis anywhere else, but less-explicit / less-narrated references include:

Itzykson and Zuber, Quantum Field Theory, section 4-1: "Quantized electromagnetic field interacting with a classical source";

Cohen-Tannoudji, Dupont-Roc, and Grynberg, Atom-Photon Interactions, exercise 17: "Equivalence between a quantum field in a coherent state and an external field", and also exercise 9.

Some related posts, from newest to oldest:

Does a photon require an EM field to exist?

How to interpret a wavepacket in quantum field theory: is it one particle or a superposition of many?

Can the equation $E = h\nu$ be used not only for light, but for radio waves? (Since they are all part of the electromagnetic spectrum)

Electromagnetic waves and photons

Photons of a radio wave

What exactly is meant by the wavelength of a photon?

Why call it a particle and not a wave pulse?

Is double slit interference due to EM/de Broglie waves? And how does this relate to quantum mechanical waves?

What is the physical nature of electromagnetic waves?

Relation between Wave equation of light and photon wave function?

Sequence of E and B field in radio waves and in single photons