 Solution

Let the incident and final photon momentum magnitudes beIn a nontrivial collinear collision, the photon is backscattered and the electron moves in the original photon direction, so momentum conservation gives electron momentum $P=p+p^{\prime }$. Conservation of relativistic energy givesSubstitute $P=p+p^{\prime }$, isolate the square root, and square. Cancellation leavesDividing by $p{p}^{\prime }$ and using the wavelength definitions gives the Compton scattering shiftThe other collinear possibility is the trivial forward solution ${\lambda }^{\prime }=\lambda$, in which the electron remains at rest.