# Compton scattering differential cross section

• Jabba64
In summary, The problem at hand is to prove an equation for the differential cross section of compton scattering of an electron and a photon. The given equation is the square of the matrix element for the process, summed over all spin states and divided by 4. Another relevant equation involves the initial energy and the size of the three momentum of the electron. The factor 1/va-vb is assumed to be 1/2. The squares of the cosines and sines in the expression for the cross section come from using the identity for these trigonometric functions. After expanding and simplifying the expression, the final equation for the differential cross section is obtained.

#### 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 .

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: