Consider an elastic collision between a photon with initial wavelength $\lambda_{0}$ moving in the $x$-direction and a stationary electron, as depicted in Fig. $34.8 b$. Use relativistic expressions for energy and momentum from Chapter 33 to show that conservation of energy and momentum yield the equations $h c / \lambda_{0}+m c^{2}=h c / \lambda+\gamma m c^{2}, h / \lambda_{0}=(h / \lambda) \cos \theta+\gamma m u \cos \phi$, and $0=(h / \lambda) \sin \theta-\gamma m u \sin \phi$, where $\lambda$ is the post-collision photon wavelength and the angles $\theta$ and $\phi$ are as shown in Fig. $34.8 b$. Solve these equations to find the Compton shift (Equation 34.8).