Center of mass energy for two relativistic colliding particles

In summary: Are you sure you have the correct equation for S? I would guess ##S = (E'_1 + E'_2)^2 - (\vec P'_1 + \vec P'_2)^2##.I am confused by the sentence starting "and then substituting". I see no substitutions.
  • #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
 
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  • #2
squareroot said:
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
squareroot said:
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.
 

Related to Center of mass energy for two relativistic colliding particles

What is the center of mass energy for two relativistic colliding particles?

The center of mass energy for two relativistic colliding particles is the total energy of the system when the particles are brought together at a common point. It takes into account the masses and velocities of both particles.

How is the center of mass energy calculated?

The center of mass energy is calculated using the formula E = √(m1^2c^4 + m2^2c^4 + 2m1m2c^2), where m1 and m2 are the masses of the two particles and c is the speed of light.

Why is the center of mass energy important in particle collisions?

The center of mass energy is important in particle collisions because it determines the maximum energy that can be produced in the collision. If the center of mass energy is high enough, it can create new particles and study their properties.

What happens if the center of mass energy is not high enough in a particle collision?

If the center of mass energy is not high enough in a particle collision, the particles will not have enough energy to create new particles. Instead, they will just scatter off of each other.

How does the center of mass energy affect the outcome of a particle collision?

The center of mass energy affects the outcome of a particle collision by determining the maximum energy available for the creation of new particles. If the center of mass energy is high enough, new particles can be created and their properties can be studied. If it is not high enough, the particles will simply scatter off of each other.

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