Collisions in special relativity

[rashida564]

Homework Statement

Collision between particles as shown in the picture below

Relevant Equations

Ei=Ef
Pi=Pf

Happy new year, I got very far in this question. I attempted to use conservation of energy first.
Ei=(Pfc)^2+mec^2 Where pf is the initial momentum of the photons
Ef=(γmec^2)+(pf'c)^2
Then used conservation of momentum
Pf=γmvcos(Θ)
Pf'=-γmvsin(Θ).
After that I added Pf and Pf'.
and used conservation of energy to find that (pf'-pf)(pf+pf)=(1-γ)me
which I substitute it's value back for pf+pf'.
simplified the expression to get that cos(Θ)-sin(Θ)=sqrt(γ-1)/sqrt(γ-1)c(Pf'-Pf)
from the question I believe that c(Pf'-Pf) should be equal to one or otherwise, I would've done algebraic mistake along the lines. But is there any reason why c(Pf'-Pf) should equal to 1.

 

[Orodruin]

Ei=(Pfc)^2+mec^2 Where pf is the initial momentum of the photons
Ef=(γmec^2)+(pf'c)^2

These equations are clearly false as the physical dimensions of the different terms are different.

 

[Orodruin]

But is there any reason why c(Pf'-Pf) should equal to 1.

It cannot be equal to one, it has physical dimension and 1 is dimensionless.