Creation of an Electron-Positron Pair by a Photon

[howin]

Homework Statement

Show that the creation of an electron-positron pair (or any particle-antiparticle pair, for that matter) by a single photon is not possible in isolation, ie, that additional mass (or radiation) must be present. (Hint: Consider the reaction at threshold, then apply conservation laws.)

Homework Equations

hc/lambda - pc = 2m(e)c^2
at threshold, hf(min) = 2m(e) c^2

The Attempt at a Solution

I know this can't work b/c you can't find a Lorentz frame in which the e+ and e- have equal and opposite momenta, b/c then the photon would be at rest which would disobey the law of special relativity.

At threshold, all the energy of the photon becomes the mass of the electron and positron so the electron and positron have KE=0.

h/lambda = p- + p+

2m(e) c = m(e) u- + m(e) u+

2c= u- + u+

 

[Meir Achuz]

I prefer doing it by showing that W^2=(E1+E2)^2-(p1+p2)^2 cannot equal zero, as it must if the two particles came from a photon.
Algebra gives W^2=2m^2+2(E1E2+p1p2 cos\theta), which can't equal zero.