How can we mathematically prove that a free electron can't absorb a photon totally?
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It relies on conservation of energy and momentum and the equation for energy in special relativity: $E^2 = (pc)^2 + (mc^2)^2$.
Here you go.
Energy of photon: $E_\gamma = \hbar\omega = p_\gamma c$, where $p_\gamma$ is the momentum of the photon. Assume the electron is initially at rest, so it's energy is simply $m_ec^2$.
By conservation of energy, the energy of the electron after it absorbs the photon is $E_\gamma + m_ec^2$, by conservation of momentum, its momentum is $p_\gamma$.
From relativity we have $E^2 = (pc)^2 + (mc^2)^2$. Subbing in all the values we calculated, we have:
\begin{equation} (E_\gamma + m_ec^2)^2=(p_\gamma c)^2+(m_ec^2)^2 \end{equation}
I will let you verify that this equation cannot hold.
Andrea
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