Center of mass energy for two relativistic colliding particles

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  • #1
squareroot
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Homework Statement:
Two particles of equal mass (m = 1GeV/c^2) collide under an angle of 30 degrees. Their momenta are 200GeV/c and 100GeV/c. What is the energy in the center of mass? How many \pi^{0} can be formed in this collision?(m_{\pi0} = 135MeV/c^2)
Relevant Equations:
p_{1} = (E_{1}, \vec{p_{1}})
p_{2} = (E_{2}, \vec{p_{2}})
in the Center of Mass frame one has : |\vec{p_{1}}| = |\vec{p_{2}}|; E_{1} = E_{2} = E

The center of mass energy is S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = 4E^2
Starting from the center of mass energy S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2})
knowing that E^2 = m_{0}c^4 + p^2*c^2 one has

S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = ( m_{0}c^4 + p_{1}^2*c^2) + m_{0}c^4 + p_{2}^2*c^2)^2 - p_{1}^2 - p_{2}^2 - 2p_{1}p_{2}cos \{theta}

and then subtituing for p_{1}, p_{2} and the angle. Could you please tell me if this approach is correct?

Thank you in advance,
sqt
 

Answers and Replies

  • #2
nrqed
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Homework Statement:: Two particles of equal mass (m = 1GeV/c^2) collide under an angle of 30 degrees. Their momenta are 200GeV/c and 100GeV/c. What is the energy in the center of mass? How many ## \pi^{0}## can be formed in this collision? ## (m_{\pi0} = 135MeV/c^2)##
Relevant Equations:: ## p_{1} = (E_{1}, \vec{p_{1}})
p_{2} = (E_{2}, \vec{p_{2}})##
in the Center of Mass frame one has : ## |\vec{p_{1}}| = |\vec{p_{2}}|; E_{1} = E_{2} = E##

The center of mass energy is ## S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = 4E^2##

Starting from the center of mass energy ## S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2})##
knowing that ## E^2 = m_{0}c^4 + p^2 c^2## one has

## S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = ( m_{0}c^4 + p_{1}^2*c^2) + m_{0}c^4 + p_{2}^2*c^2)^2 - p_{1}^2 - p_{2}^2 - 2p_{1}p_{2}cos \{theta} ##

and then subtituing for## p_{1}, p_{2}## and the angle. Could you please tell me if this approach is correct?

Thank you in advance,
sqt
I corrected your post so that the LaTex would show. For some reason I can only edit part of your post. If you reply to my message, you will see how I made the LaTeX show up.
 
  • #3
PeroK
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Homework Statement:: Two particles of equal mass (m = 1GeV/c^2) collide under an angle of 30 degrees. Their momenta are 200GeV/c and 100GeV/c. What is the energy in the center of mass? How many \pi^{0} can be formed in this collision?(m_{\pi0} = 135MeV/c^2)
Relevant Equations:: p_{1} = (E_{1}, \vec{p_{1}})
p_{2} = (E_{2}, \vec{p_{2}})
in the Center of Mass frame one has : |\vec{p_{1}}| = |\vec{p_{2}}|; E_{1} = E_{2} = E

The center of mass energy is S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = 4E^2

Starting from the center of mass energy S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2})
knowing that E^2 = m_{0}c^4 + p^2*c^2 one has

S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = ( m_{0}c^4 + p_{1}^2*c^2) + m_{0}c^4 + p_{2}^2*c^2)^2 - p_{1}^2 - p_{2}^2 - 2p_{1}p_{2}cos \{theta}

and then subtituing for p_{1}, p_{2} and the angle. Could you please tell me if this approach is correct?

Thank you in advance,
sqt
I think you need more details to see what you are doing. In general, you need to distinguish between quantities in different frames. E.g. use ##E', \vec P' = \vec p'_1 + \vec p'_2## for quantities in the CoM frame.
 

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