Additional Comments on Midterm Problems

1. Assume that the neutrino burst of SN1987A was instantaneous, i.e. all neutrinos were emitted at once. This assumption is not realistic, but the duration of the burst is believed to be only of a few seconds and a more realistic analysis does not change the result very much.
1. The angles listed in the table in Kamiokande paper are the angles between the positron direction and the SN1987A direction. For instance, a positron which goes with the same direction as the original neutrino would have the angle 180 degrees.
1. The unit "pc" (parsec) is a unit for distance used often in astronomy. Look up the table "Astrophysical Constants" in the booklet. 1pc is defined as a distance for which the apparent direction changes by 2 seconds (1/3600 degree) between summers and winters.
1. Once you know the energy of each neutrino, you can calculate how much time it took to reach the Earth as a function of its mass. And the arrival times shouldn't differ more than the time spread of the burst observed in Kamiokande. Then you find that the neutrino mass must be less than something.
1. The reaction anti-electron-neutrino+proton -> positron+neutron can be understood by the following mechanism. The incoming anti-electron-neutrino emits a W- boson and converts to a positron. This W- is absorbed by one of the up quarks inside proton and it becomes a down quark. Then the proton becomes a neutron. This information, however, is not needed for solving the problem.
3. Do not think about quark contents of the rho, omega, and pi mesons. Just discuss wave functions of mesons and the conserved quantum numbers. Note that the decays discussed here are all caused by strong interactions. The analysis in this problem will be similar to the construction of deterium wave function we did in the class. Don't bother trying to figure out the radial wave function. It depends on the details of the interactions. The point of the problem is that just the conservation of basic quantum numbers is enough to give you interesting information on the decay modes. Think only about the isospin, spin, relative orbital angular momentum and statistics (Bose vs Fermi).
3. For a two-body system, you can always separate the center-of-gravity motion (to be more precise, center-of-energy motion in the relativistic case) and the relative motion. The "relative orbital angular momentum" refers to the orbital angular momentum for the relative coordinate x1-x2. For the initial state of rho or omega meson, the only angular momentum, after you separate out the center-of-energy motion, is its spin. Therefore, the conservation of the angular momentum requires that the spin of the parent particle is the same as the sum of the relative orbital angular momentum and spins of the daughter particles.
4. In (f) and (g), what I meant was I3|1/2,1/2>=(1/2)|1/2,1/2> etc. The state was missing in r.h.s of equations. J3 is the same as Jz.
4. The unit for the proton magnetic moment in the booklet is given in the table "Physical Constants" as the "nuclear magneton."
A correspondence with Nathan
A correspondence with Gary
Another correspondence with Gary
A correspondence with Taro
A correspondence with Manuel

Good luck!

[email protected]