High-energy physics: momentum transfer

[Chiborino]

Homework Statement

I have an electron of 20 GeV and negligible mass that collides with a stationary proton (mc^2 = 9.38 GeV) and deflects at an angle of 5°. I'm asked to find the square of the four-momentum transfer, q²

Homework Equations

q = P - P', where P/P' is a 4-momentum vector <p_x, p_y, p_z, iE>
a "primed" quantity represents a value after the collision with the proton.

The Attempt at a Solution

I took the quantity P-P' and squared it:
q² = (P-P')*(P-P') = P² + P'² -2P*P'
I'm told the first two terms are negligible due to the electron's mass being negligible, but I'm not sure I see the sense in that.
Anyways, continuing on, I then have:
q² = -2P*P' = -2(p_x*p_x' + p_y*p_y'+p_z*p_z' -E*E')
or q² = -2p*p' + 2E*E'
This is where I'm stuck. Should I also assume the product of the 3-momenta is 0 and carry on with the +2EE' term I'm left with? And what do I even do about the E' since I don't know that quantity?

 

[vela]

Homework Statement

I have an electron of 20 GeV and negligible mass that collides with a stationary proton (mc^2 = 9.38 GeV) and deflects at an angle of 5°. I'm asked to find the square of the four-momentum transfer, q²

That's a pretty massive proton!

Homework Equations

q = P - P', where P/P' is a 4-momentum vector <p_x, p_y, p_z, iE>
a "primed" quantity represents a value after the collision with the proton.

The Attempt at a Solution

I took the quantity P-P' and squared it:
q² = (P-P')*(P-P') = P² + P'² -2P*P'
I'm told the first two terms are negligible due to the electron's mass being negligible, but I'm not sure I see the sense in that.
Anyways, continuing on, I then have:
q² = -2P*P' = -2(p_x*p_x' + p_y*p_y'+p_z*p_z' -E*E')
or q² = -2p*p' + 2E*E'
This is where I'm stuck. Should I also assume the product of the 3-momenta is 0 and carry on with the +2EE' term I'm left with? And what do I even do about the E' since I don't know that quantity?

With the small deflection, you would expect that the electron didn't give up much of its energy, so you wouldn't expect the product of the three-momenta to vanish. Unfortunately, you're not going to be able to just erase terms you don't want to deal with.

You need to use conservation of energy and momentum to figure out $\vec{p}'$ and E'. Note that because the electron is essentially massless in this problem, the situation is essentially the same as Compton scattering.