User:Falcorian/Pair Production

Conservation of Momentum
Why couldn't the newly created positron and electron absorb the momentum of the photon? How would the conservation law break without a nucleus? The article doesn't give a reason for this, just states this as a fact. Could someone shed some light on this? --PAStheLoD 02:54, 8 June 2010 (UTC)


 * The equations don't work out (or at least the answers aren't real). For example, define $$E_{\gamma} = 2 m c^2 + \epsilon$$. Then:


 * Energy: $$E_{\gamma} = 2 m c^2 + \epsilon = 2 m c^2 + \frac{1}{2} m ( v_{e^-}^2 + v_{e^+}^2 )$$


 * Momentum: $$ \frac{E_{\gamma}}{c} = \frac{2 m c^2 + \epsilon}{c} =m ( v_{e^-} + v_{e^+} ) $$


 * Combining: $$\frac{2 m c^2 + \frac{1}{2} m ( v_{e^-}^2 + v_{e^+}^2 )}{c} = m ( v_{e^-} + v_{e^+} )$$


 * Simplify: $$4 + \left (\frac{v_{e^-}}{c}\right )^2 + \left (\frac{v_{e^+}}{c}\right )^2 = 2 \left ( \frac{v_{e^-}}{c} + \frac{v_{e^+}}{c}  \right )  $$


 * Solve for either v: $$ v_{e^-} = c \left( 1 \pm \sqrt{-3 + 2  \frac{v_{e^+}}{c} - \left ( \frac{v_{e^+}}{c}\right)^2} \right)$$


 * Which is complex.