= Solution
Let the incident and final photon momentum magnitudes be
$$
p=\frac h\lambda,
\qquad
p'=\frac h{\lambda'}.
$$
In a nontrivial collinear collision, the photon is backscattered and the electron moves in the original photon direction, so momentum conservation gives electron momentum $P=p+p'$. <Conservation of relativistic energy> gives
$$
pc+mc^2=p'c+\sqrt{m^2c^4+P^2c^2}.
$$
Substitute $P=p+p'$, isolate the square root, and square. Cancellation leaves
$$
mc(p-p')=2pp'.
$$
Dividing by $pp'$ and using the wavelength definitions gives the <Compton scattering> shift
$$
\boxed{\lambda'=\lambda+\frac{2h}{mc}}.
$$
The other collinear possibility is the trivial forward solution $\lambda'=\lambda$, in which the electron remains at rest.
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