Two (fictional) heavy particles are fired towards each other in a linear accelerator. Particle 1, moving towards the right, has a rest mass m1 = 16mp (mp is the proton mass) and velocity v1 = 3c/5 in the laboratory frame. Particle 2, moving towards the left, has a rest mass m2 = 9mp. The particles collide elastically and form a metastable particle, stationary in the laboratory frame.
(a) Show, using conservation of total momentum before and after the collision, that the velocity of particle 2 in the laboratory frame is given by v2 = 4c/5.
(b) What is the mass of the metastable particle after the collision, in units of proton mass? Is mass conserved in the collision?
Now consider a reference frame co-moving with particle 1.
(c) In this reference frame, what is the velocity of particle 2?
(d) The rest mass is Lorentz invariant, and therefore independent of the choice of reference frame. Use conservation of energy and momentum in this new reference frame to prove that the collision will result in a particle of the same mass given by your answer to part (b) above, and moving to the left with velocity v1.
After some period of time, the metastable particle decays into a shower of lighter particles, including protons, electrons, neutrinos and photons.
(e) According to quantum mechanics, the shower of particles will scatter in all directions with equal probability. Is it possible to determine in advance the trajectory of any individual particle? Why?
(f) Several of the particles pass through a double-slit positioned near the point of the collision. Describe what happens as individual particles pass through it.
(g) Briefly describe how the resulting interference pattern will differ for the various types of particles passing through: protons, electrons, neutrinos (mass close to zero) and photons. Assume that the distribution of particle energies is the same for each kind of particle.