Write down the 4-momentum of a particle with energy $E$ and 3-momentum p. State the relationship between the energy $E$ and wavelength $\lambda$ of a photon.

An electron of mass $m$ is at rest at the origin of the laboratory frame: write down its 4 -momentum. The electron is scattered by a photon of wavelength $\lambda_{1}$ travelling along the $x$-axis: write down the initial 4-momentum of the photon. Afterwards, the photon has wavelength $\lambda_{2}$ and has been deflected through an angle $\theta$. Show that

$$\lambda_{2}-\lambda_{1}=\frac{2 h}{m c} \sin ^{2}\left(\frac{1}{2} \theta\right)$$

where $c$ is the speed of light and $h$ is Planck's constant.