Calculate relativistic com frame for two particles?

[jason12345]

Does anyone know of a standard way of calculating the com frame velocity for two particles moving at arbitary velocities in the lab frame?

It's strange that this standard result isn't even in Goldstein's et al book

 

[Vanadium 50]

E² - p² = m²

βγ = p/m.

 

[jtbell]

The velocity of a single particle in terms of its energy and momentum is given by

$$\beta = \frac {pc}{E}$$

Given this, what would you expect the velocity of the "equivalent particle" representing the motion of a system of two particles to be? Or indeed, any number of particles?

You can get the result a bit more rigorously by using the Lorentz transformation for energy and momentum

$$p^{\prime} c = \gamma (pc - \beta E)$$

and requiring that the total momentum in the primed frame be zero, i.e. for a system of two particles $p_1^{\prime} c + p_2^{\prime} c = 0$.

 

[Meir Achuz]

Calulate $${\bf P=p_1+p_2)}$$, and $$E=E_1+E_2$$.
Then $${\bf V=P}/E.$$.

 

[jason12345]

@itbell and Meir Achz, yes I see what you mean!

The key for me was visualising the disintegration of a particle into p1 and p2, while using conserving 4-momentum.

This really is a very elegant, beautiful result, which I can't find anywhere in my copy of Goldstein, 3rd edition, nor in any of the problems. Maybe it's mentioned in books devoted to the dynamics of particle collisions.

 

[jason12345]

Calulate $${\bf P=p_1+p_2)}$$, and $$E=E_1+E_2$$.
Then $${\bf V=P}/E.$$.

For c=1, but in general $${\bf V=P}c^2/E.$$

yes?

 

[Bob S]

The key for me was visualising the disintegration of a particle into p1 and p2, while using conserving 4-momentum.

This really is a very elegant, beautiful result, which I can't find anywhere in my copy of Goldstein, 3rd edition, nor in any of the problems. Maybe it's mentioned in books devoted to the dynamics of particle collisions.

Much of the formalism, including the concept of invariant mass during particle disintegration, is included here:
http://pdg.lbl.gov/2010/reviews/rpp2010-rev-kinematics.pdf

 

[Meir Achuz]

For c=1, but in general $${\bf V=P}c^2/E.$$

yes?

I use light years.

 

[BenHastings]

So in Fig 39.3 it shows the constraints of the final state.

What determines in what final state the system of equations will stabilize?

 

[jtbell]

The final state is located randomly somewhere inside the shaded area, with a probability distribution that is determined by the matrix element for the process that we're dealing with.

It's rather like asking "what determines exactly when a particular radioactive nucleus will decay?"

 

[BenHastings]

So this is where deterministic measurement meets the road to philosophical physics.

This is where "squeezing" occurs?