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The problem is asking me to find the minimum photon energy that would produce an electron-positron pair when it collides with a free electron at rest.

This is my attempt at trying to conserve both energy and momentum:

Energy before: $hf_{min} + m_ec^2$

Energy after: $2m_ec^2 +E_e$

Where $E_e$ is the total energy of the free electron after collision which is

$E_e = \sqrt{m_e^2 c^4 + p^2 c^2}$

So the equation for conservation of energy is:

$hf_{min} + m_ec^2 = 2m_ec^2 + \sqrt{m_e^2 c^4 + p^2 c^2}$

$hf_{min} = m_ec^2 + \sqrt{m_e^2 c^4 + p^2 c^2}$

I'm assuming the pair is produced with zero momentum, now I conserve momentum by letting the momentum of the photon before collision equal the momentum of the free electron after collision, so:

$\frac{hf_{min}}{c} = p$ , I then substitute this into the energy conservation equation

$hf_{min} = m_ec^2 + \sqrt{m_e^2 c^4 + h^2 f_{min}^2}$

Now this is impossible, what went wrong with my assumptions?

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