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Suppose we have some interaction between a photon and an (initially) stationary electron, and we wished to find the final momentum of the electron. Should we solve this using conservation of momentum, using the relationship $$\mathbf{p}_{\text{photon-initial}}=\mathbf{p}_{\text{photon-final}}+\mathbf{p}_{\text{electron-final}}$$ or by conservation of energy, using the relationship \begin{align} E_{\text{photon-initial}}+E_{\text{electron-initial}}=E_{\text{photon-final}}+E_{\text{electron-final}}\\ \Longrightarrow m_ec^2-\Delta E_{\text{photon}}=\sqrt{p^2c^2+m_e^2c^4}, \end{align} and then solve for momentum?

I do get different answers for specific problems regarding photon and stationary electron interactions, and I am wondering if it is a matter of my calculations being off, or some other fundamental error.

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You should remember that photons don't have any mass, so relation is $E=p/c$ for photons. Moreover, you have to count electrons mass when you apply energy equations. Electron mass will vanish, but it's important to take account of it. I mean, before the square root, you don't have the electron mass, you have the kinetic energy.

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