How Do You Calculate Proton Beam Energy in a Lab Frame?

[FilipLand]

Homework Statement

a) [Solved] What is the threshold energy in the center of mass for production of an antiproton in a proton-proton collision? Make sure quantum numbers are conserved. (This one was easy and I manage to solve it)

b) In a fixed target experiment calculate the proton beam energy corresponding to this threshold and the energy of the produced anti-proton in this laboratory frame.

Homework Equations

The Attempt at a Solution

On b) is where I'm a little stuck. On a) I simply squared the sum of the four-momenta and putted equal to (E_cm)^2 / c^2. Where E_cm = 3mc^2

But when I get an energy, how do I convert that to the lab-frame?

 

[Orodruin]

What stops you from doing the same computation as in (a) but using the expressions for the 4-momenta in the lab frame? These 4-momenta contain the quantities that you are looking for, such as the proton beam energy.

 

[FilipLand]

What stops you from doing the same computation as in (a) but using the expressions for the 4-momenta in the lab frame? These 4-momenta contain the quantities that you are looking for, such as the proton beam energy.

I'm not certain how to express the four-momenta in any other way, nor in lab-frame. How would you put up the equations?

 

[Orodruin]

What quantities do you have available in the lab frame? How does the 4-momentum look in an arbitrary frame?

 

[FilipLand]

What quantities do you have available in the lab frame? How does the 4-momentum look in an arbitrary frame?

Other then the general form p=(E/c, Px, Py, Pz) I can't tell, or the quantities other then masses.

 

[Orodruin]

So, in a fixed target experiment, where one of the protons are at rest, what are the 4-momenta of the two protons in terms of the proton mass and the energy of the moving proton?

 

[FilipLand]

So, in a fixed target experiment, where one of the protons are at rest, what are the 4-momenta of the two protons in terms of the proton mass and the energy of the moving proton?

p_1=(E_1 /c, 0, 0, sqrt(p^2c^2 - m^2 c^4)) and p_2=(E_2, 0, 0, 0)

But honestly, this way of guidance makes me nothing but more confused. So no more of these hints please.

 

[Orodruin]

p_1=(E_1 /c, 0, 0, sqrt(p^2c^2 - m^2 c^4)) and p_2=(E_2, 0, 0, 0)

This is not correct. In particular the first. The second you have additional information that you are not taking into account: what is the rest energy of a proton? Later, you will want to make sure that the centre of mass energy is sufficient to produce the additional particle. How would you express that using the 4-momenta? Note that the inner products are Lorentz invariant and can be computed in any frame. In particular, if you are interested in the quantities in the lab frame, you should compute the inner products in the lab frame.

But honestly, this way of guidance makes me nothing but more confused. So no more of these hints please.

Sorry, but this attitude will get you nowhere. Did you happen to read the homework guidelines? Nobody will solve the problem for you. Also, you would get more out of it faster if you checked in more than once per day.