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## Homework Statement

Two photons collide and create an electron-positron pair:

y

_{1}+y

_{2}-> e

^{-}+e

^{+}

If the wavelength of y

_{1}is 1 mm calculate the energy threshold for photon y

_{1}to produce the electron-positron pair.

Suppose E(y

_{1})=hc/[itex]\lambda[/itex]

_{2}where [itex]\lambda[/itex]

_{2}=1.1 mm

Hints: In the center of mass frame of reference, in which the total momentum is 0, the threshold reaction would just produce the two particles at rest. Then in the frame with photon energies E1

and E2 , the two particles would have the same momentum.

## Homework Equations

Given:

Electron Energy=(p

^{2}c

^{2}+m

^{2}c

^{4})

^{1/2}

Photon Energy=pc where p=momentum

## The Attempt at a Solution

The energy of y

_{2}is planck's constant*c/1.1 mm = 1.81*10

^{-22}Joules

I'm a bit confused about where to go from here. Seeing as how energy and momentum are conserved the total energy of y

_{1}and y

_{2}will have to equal the energy of the electron-positron pair. Since I have E(y

_{2}) and need to find E(y

_{1}) I need to find the energy of the electron-positron pair.

I'm given the energy equation for the electron and their mass, c, and p should all be equal so their energies will be equal.

Is the following assumption correct?

E(y

_{2})=2*(p

^{2}c

^{2}+m

^{2}c

^{4})

^{1/2}-E(y

_{1})

The problem is I don't have a wavelength to figure the energy of the resulting electron-positron pair.

Any suggestions?

Thank you!