Answer
See answers.
Work Step by Step
a. Start in a frame of reference where the isolated electron is at rest, i.e., it has zero KE and zero momentum.
If it emits only a single photon, that photon would have momentum of E/c. By conservation of momentum, the electron must recoil in the opposite direction with an equal but opposite momentum. However, it would then be moving, so its total energy after the emission (electron’s rest energy, plus KE, plus photon energy) would be greater than the total energy before (just the electron’s rest energy). By conservation of energy, the proposed reaction is impossible.
b. Consider the photon exchange in figure 32–8. The exchange is possible because of the Heisenberg uncertainty principle. The photon exists for such a short time $\Delta t$ that energy conservation can be violated by an amount $\Delta E$.
In other words, a process can borrow an energy $\Delta E$, violating conservation of energy, if the violation happens over a short enough duration $\Delta t$ that $\Delta E\Delta t \approx \hbar$.
