Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 3 / 10A (Special Relativity)

A photon of wavelength $\lambda$ has energy $E=h\nu =hc/\lambda$ and momentum $\mathbf{p}=(h/\lambda )\hat{\mathbf{n}}$ along its propagation direction.

The specified Compton scattering formula requires the electron to be initially at rest in the frame where the wavelengths and angle are measured. This initial-rest assumption is implicit in the printed request; it is not valid for arbitrary initial electron momentum. Use four-momentum with Minkowski metric signature $(+---)$: initially $P=(mc,0)$, while $K=(E/c,(E/c)\hat{\mathbf{n}})$ and $K^{\prime }=(E^{\prime }/c,(E^{\prime }/c){\hat{\mathbf{n}}}^{\prime })$ are null. Conservation gives $P^{\prime }=P+K-K^{\prime }$, and the outgoing electron satisfies the same mass-shell condition $P^{\prime 2}=P^2=m^2{c}^2$. HenceAfter dividing by $E{E}^{\prime }$ and substituting the photon energy relation,The scale $h/(mc)$ is the electron Compton wavelength. For a moving initial electron, the earlier invariant identity remains valid, but $P\cdot K$ includes its initial velocity and the wavelength shift is different.