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:

$\psi _{i}(\mathbf {r} )={\frac {e^{i\mathbf {k_{i}} \cdot \mathbf {r} }}{V^{3/2}}}$

$\psi _{i}(\mathbf {r} )={\frac {e^{i\mathbf {k_{f}} \cdot \mathbf {r} }}{V^{3/2}}}$

The transition rates are:

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

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

where,

V =