(a) A photon with energy $E_{1}$ in the laboratory frame collides with an electron of rest mass $m$ that is initially at rest in the laboratory frame. As a result of the collision the photon is deflected through an angle $\theta$ as measured in the laboratory frame and its energy changes to $E_{2}$.

Derive an expression for $\frac{1}{E_{2}}-\frac{1}{E_{1}}$ in terms of $\theta, m$ and $c$.

(b) A deuterium atom with rest mass $m_{1}$ and energy $E_{1}$ in the laboratory frame collides with another deuterium atom that is initially at rest in the laboratory frame. The result of this collision is a proton of rest mass $m_{2}$ and energy $E_{2}$, and a tritium atom of rest mass $m_{3}$. Show that, if the proton is emitted perpendicular to the incoming trajectory of the deuterium atom as measured in the laboratory frame, then

$$m_{3}^{2}=m_{2}^{2}+2\left(m_{1}+\frac{E_{1}}{c^{2}}\right)\left(m_{1}-\frac{E_{2}}{c^{2}}\right)$$