Question

Problem 4. Electron-positron production An electron-positron pair (positron is electron’s antiparticle, it has the same mass as electron, but opposite charge) can be produced what two photon are collided. Two photons of frequency ω are collided head-on. What will be the electron’s momentum? Electron’s rest mass is me.

Answer 1

So the process is shown below, ♡ é et (photon) [phaten) (elee toen) (ponti tron) Let E, and Eq are the energien of election and position respectively. Then 5,? = $? + mely 3 where per and bet Per met an the momentums.of electron and position respectively. Hence, E- Ebel-being => [E,+ Ex) (6,-E,) pepe and a 2 ar Equeal. Thus Again, the frequenues of the photom from Energy conservation law, we have aħw E, TE, 42 mec? → (6 +£;) 2[*w - mec?) - @ Therefore, from-, we get E,-€ = (tiro- mee? po 2 Since the phatons have some frequency then their Wavelengtha aho be equal. Therefore, the momentum of the combination of two photons, travelling along the opposite direction is 2010, and according to the momentum conservation law, the momentum combi- anation of the electron position system, in also zero. Therefore, Pe- + Þet -0.= Then from – ③ we have E,-E2 From the combination of equation- and , we have, 2E, = sakkw - mec?) Ei= tw-mec2 Hence, the momentom of electron e's E,?- me?cy hen twmit" - Rtwmee? -matico 2 2 62² tw² - 2ħw mec? ke = √2²_2kw mee? momentum of eletron

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