Calculating Max/Min Energies of Particle P1 in Decay of A

[crit]

Particle A, at rest, decays into three or more particles: P1, P2, ..., Pn.
Determine the maximum and minimum energies that P1 can have in such a decay, in terms of various masses.

My solution:

First of all, the decay should not occur if the rest mass of A would be smaller than the sum of the rest mass of P1, P2, ..., Pn.

energy conservation. E1 minimum is the rest energy of P1 which is just created, with zero momentum. The rest of the energy is taken by the rest energies and momentum of P2, P3, etc.

E1 max if E2+E3+...En is minimum, that is if P2 is at rest, P3 is at rest, ..., Pn is at rest. But this is false, because then, by conservation of momentum, P1 should also be at rest and would have E1 min, not maximum. At least two particles should have non zero momentum. E1 and another one. But then? I tried an analytical approach, but the calculus would need Mathematica.

Any suggestions for a straighforward solution? Thanks.

 

[dextercioby]

The first part is okay.Indeed,simple logics and virtually no computations will lead to the result that the minimum energy of the particle is the rest one.

For the second...I'm not really sure it is a min/max problem,one which would require Lagrange multipliers...It's more about logics.

Indeed,the same logics tells u that at least 2 particles must have nonzero momentum.The logics tells u that this number MUST BE EXACTLY 2.I hope u see why...If there are at least three,then the energy of the one u're interested in will be smaller.(the sum of the three would be constant...)
My logics says that form the N-1 other particles,the one with the smallest rest mass will fulfil the requirement.Let's label it with "2" and the particle of interest with "1".Think about it.The energies of the 2 particles are constant and equal to the difference between the initial particle's rest energy ($m_{A}$) and the sum of the rest energies of the other (N-2) particles...

So
$$E_{1}+E_{2}=constant=m_{A}-\sum_{j=3}^{n} m_{j}$$(1)

Okay??
But u know that the 2 particles must have nonzero momenta,however,of opposite sign and same modulus (this,as u may have noticed,results from the conservation of total momentum in the decay process)

$$E_{1}=\sqrt{\vec{p}_{1}^{2}+m_{1}^{2}}$$(2)
$$E_{2}=\sqrt{\vec{p}_{2}^{2}+m_{2}^{2}}$$(3)

$$\vec{p}_{1}+\vec{p}_{2}=\vec{0}$$ (4)


From (4) u find $\vec{p}_{2}$ in terms of $\vec{p}_{1}$ and then take square and substitute in (3).Use the relation (2),the "new version" of (3) and the condition (1) to get one ugly equation in terms of $\vec{p}_{1}^{2}$ .

Then substitute in (2) and find the maximum energy...

Daniel.

P.S.Bai,Adita,ce mama ma-sii,nu putea sa-ti imaginezi si tu chestia asta??Si cu "crit-clit" ce e??
P.P.S.Asta e mai simpla decit o problema de la Judeteana intr-a 12-a...

 

[crit]

I have discovered the same reasoning as yours during my last class and I have calculated that momentum of both P2 and P3. But I want mathematical result to tell me that P3 with be the one with the smallest rest mass and not "logic", which in this case is just an intuition (that was also my intuition). I am annalizing that and I will write again here.

 

[dextercioby]

What P_{3} are u talking about??I didn't mention any P_{3}...I have chosen P_{2} as the momentum of the particle with smallest rest mass...The charcateristics of a third (why not a 4-th,5-th and so on) particle do not interviene whatsoever...

I don't know what u find against logics...Mathematics is founded on logics...

Waiting for an answer and "nu-mi mai trimite meiluri !" .Sint tot timpul online.

Daniel.