View source for Radiation Interacting with Free Electrons Problem

== Queston ==

Show that free electrons can neither emit nor absorb photons.

== Solution ==

Let us initialy assume that it is posible for free electrons to emit and absorb photons. We know that the wave funtion for a free electorn is a plane wave, so our initial and final wave functions are:

<math> \psi_{i}(\mathbf{r}) = \frac{ e^{i \mathbf{k_{i}} \cdot \mathbf{r}} }{ V^{3/2} }</math>

<math> \psi_{i}(\mathbf{r}) = \frac{ e^{i \mathbf{k_{f}} \cdot \mathbf{r}} }{ V^{3/2} }</math>

The transition rates are:

<math> \Gamma^{abs}_{i \rightarrow f} = \frac{2 \pi}{\hbar}|<\psi_{f}|V^{abs}|\psi_{i}>|^{2}\delta(E_{f} - E_{i} - \hbar\omega) </math>

<math> \Gamma^{ems}_{i \rightarrow f} = \frac{2 \pi}{\hbar}|<\psi_{f}|V^{ems}|\psi_{i}>|^{2}\delta(E_{f} - E_{i} - \hbar\omega) </math>


where,

V =