Compton scattering differential cross section

[Jabba64]

1. The problem statement, all variables and given/known

I have to prove an equation for the differential cross section of compton scattering of an electron and a photon (electron (P) + photon(K) ⇒ electron(P') + photon (K') ) where P and so on are the inital and final four momenta.

Given is that the square of the matrix element for the process, summed over all spin states and divided by 4 is given by:

i have to prove:

Where E and p are the initial energy and the size of the three momentum of the electron.
We can also use the following formula:

Where i believe we assume the factor 1/va-vb to be 1/2.

Can someone help me solve this? I also have no idea where the squares of the cosines and sines in the expression for the cross section come from .

 

[Asim Riaz]

2. Relevant equationsThe square of the matrix element for the process, summed over all spin states and divided by 4 is given by: Where E and p are the initial energy and the size of the three momentum of the electron.We can also use the following formula: Where i believe we assume the factor 1/va-vb to be 1/2.3. The attempt at a solutionI have started by expanding the numerator using the formula for the momentum transformation: Then I expanded the denominator in a similar way: After that I used the identity for the cosines and sines to get the squares of them: Finally I expanded the expression and simplified it with some basic algebraic manipulations to get the desired result: So the final equation for the differential cross section of compton scattering of an electron and a photon is: