Free electron can't absorb a photon

Why can't a free electron absorb a photon? But a one attached to an atom can.. Can you explain to me logically and by easy equations? Thank you..

Answer

Can't believe I can't find a duplicate. It is because energy and momentum cannot be simultaneously conserved if a free electron were to absorb a photon. If the electron is bound to an atom then the atom itself is able to act as a third body repository of energy and momentum.

Details below:

Conservation of momentum when a photon ($\nu$) interacts with a free electron, assuming that it were absorbed, gives us $$\begin{equation} p_{1} + p_{\nu} = p_{2}, \tag{1} \end{equation}$$ where $p_1$ and $p_2$ are the momentum of the electron before and after the interaction. Conservation of energy gives us $$\begin{equation} \sqrt{(p_{1}^{2}c^{2} + m_{e}^{2}c^{4})} + p_{\nu}c = \sqrt{(p_{2}^{2}c^{2}+m_{e}^{2}c^{4})} \tag{2} \end{equation}$$ Squaring equation (2) and substituting for $p_{\nu}$ from equation (1), we have $$p_{1}^{2}c^{2} + m_{e}^{2}c^{4} + 2(p_{2}-p_{1})\sqrt{(p_{1}^{2}c^{2}+m_{e}^{2}c^{4})} + (p_{2}-p_{1})^{2}c^{2}=p_{2}^{2}c^{2}+m_{e}^{2}c^{4}$$ $$(p_{2}-p_{1})^{2}c^{2} - (p_{2}^{2}-p_{1}^{2})c^{2} + 2(p_{2}-p_{1})c\sqrt{(p_{1}^{2}c^{2}+m_{e}^{2}c^{4})} = 0$$ Clearly $p_{2}-p_{1}=0$ is a solution to this equation, but cannot be possible if the photon has non-zero momentum. Dividing through by this solution we are left with $$\sqrt{(p_{1}^{2}c^{2}+m_{e}^{2}c^{4})} - p_{1}c =0$$ This solution is also impossible if the electron has non-zero rest mass (which it does). We conclude therefore that a free electron cannot absorb a photon because energy and momentum cannot simultaneously be conserved.

NB: The above demonstration assumes a linear interaction. In general $\vec{p_{\nu}}$, $\vec{p_1}$ and $\vec{p_2}$ would not be aligned. However you can always transform to a frame of reference where the electron is initially at rest so that $\vec{p_1}=0$ and then the momentum of the photon and the momentum of the electron after the interaction would have to be equal. This then leads to the nonsensical result that either $p_2=0$ or $m_e c^2 = 0$. This is probably a more elegant proof.