The process

In 1923 Arthur Compton shone X-rays at graphite and measured the wavelength of what came back. It came back longer — and by an amount that depended only on the angle:

\lambda' - \lambda = \lambda_C\,(1 - \cos\theta), \qquad \lambda_C = \frac{h}{m_e c}

Light bouncing off a free electron loses energy, like a billiard ball. That result did more to establish that photons carry momentum than any argument about the photoelectric effect, and it is pure energy–momentum conservation — no field theory anywhere in it.

Named confusinglyThree different things are called Compton: an effect, a process, and a length.

Often said

Compton scattering is the Compton effect — a photon bounces off an electron and comes back longer, by the Compton wavelength.

Actually

Three distinct objects share the name, and they are not even the same kind of object.

The Compton effect is the 1923 result $\lambda' - \lambda = \lambda_C(1-\cos\theta)$ . It is pure energy–momentum conservation — no field theory, no cross section. It says where the light goes.
Compton scattering, in this course, is the QED process $e^-\gamma \to e^-\gamma$ , whose observable is a cross section: how much light goes where. That is the Klein–Nishina formula, and it needs the Dirac equation.
The Compton wavelength $\lambda_C = h/m_ec \approx 2.43$ pm is a fixed length set by the electron mass. It is the coefficient in the shift formula, not the shift — the actual shift runs from zero (forward) to $2\lambda_C$ (backward).

Why it’s natural

All three came out of the same experiment and are introduced on the same page of most treatments, so nothing signals that one is a kinematic identity, one a dynamical prediction, and one a constant. The shift formula makes it worse by putting the constant and the effect in a single equation, which invites reading $\lambda_C$ as "the amount it shifts". There is also a concrete trap for anyone searching a text rather than reading it: Tong's notes contain the string "Compton" only in "Compton wavelength" and never compute the process at all, so a reader looking for the scattering calculation finds the length scale and reasonably concludes the topic is covered.

Srednicki — Problem 11.2 states the Klein–Nishina formula, the dynamical result; section 59 computes the process it belongs to.
Tong — Search these notes for "Compton" and every hit is "Compton wavelength" — the process itself is never computed, which is the trap in one line.

All subtleties in this course →

But notice what it does not tell you. It says where the light goes. It says nothing about how much goes there. Point your X-rays at the graphite and the shift formula is silent on whether you should expect a bright forward beam and a faint backward one, or an even glow in all directions.

Answering how much is a cross section, and a cross section needs an amplitude. That is this lesson: $e^-\gamma \to e^-\gamma$ , at tree level, in QED.

It is also the first process in this course with a photon on an external leg. Every amplitude you have squared so far — $\mu^+\mu^-$ , Bhabha, Møller — had fermions on all four legs, and squaring meant spin sums and nothing else. A photon on the outside brings a polarization vector with it, and summing over polarizations turns out to be considerably more delicate than summing over spins.

Two of the four legs are massless here, so the Mandelstam identity you met in lesson 2.2 reads

s + t + u = 2m_e^2

and we will work, as always, in the centre-of-mass frame.