\begin{problem*}{28.16} % Compton effect
A $0.110\U{nm}$ photon collides with a stationary electron.  After
the collision, the electron moves forward and the photon recoils
backward.  Find the momentum and the kinetic energy of the electron.
\end{problem*}

\begin{solution}
The photon scatters by $180\dg$, so from the Compton shift equation
\begin{equation}
  \lambda' = \lambda_0 + \frac{h}{m_e c}(1-\cos 180\dg)
    = \lambda_0 + \frac{2h}{m_e c}
    = (110+4.85)\U{pm} = 115\U{pm} \;.
\end{equation}
The kinetic energy of the electron is given by the change in photon
energy (just like problem 28.15).
\begin{equation}
  K = E_0 - E' = \frac{hc}{\lambda_0} - \frac{hc}{\lambda'}
    = 11.3\U{keV} - 10.8\U{keV}
    = \ans{476\U{eV}} \;.
\end{equation}
We conserve momentum to find the electron's momentum, using
$p_\text{photon}=E/c$.
\begin{align}
  p_i &= \frac{E_0}{c} = p_f = p_e - \frac{E'}{c} \\
  p_e &= \frac{E_0+E'}{c} = (11.3+10.8)\U{keV/$c$} = \ans{22.1\U{keV/$c$}}
    = \ans{1.18\E{-23}\U{kg$\cdot$m/s}} \;.
\end{align}
\end{solution}
