= Solution
After thermalization, approximate the <electron> and <Positron> as initially at rest in their <centre-of-momentum frame>. Their total <energy> is $2m_ec^2$ and total <momentum> is zero. Two <photons> must therefore have equal <energies> and opposite <momenta>, since $|\mathbf p_\gamma|=E_\gamma/c$. <Conservation of energy> gives
$$
2E_\gamma=2m_ec^2,\qquad \boxed{E_\gamma=m_ec^2=511\,\mathrm{keV}.}
$$
Residual pair <momentum> and electron binding cause small departures from exact collinearity and exact photon <energy>; the standard <PET> calculation neglects them.
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