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 antielectronneutrino+proton >
positron+neutron can be understood by the following mechanism. The
incoming antielectronneutrino 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 twobody system, you can always separate the
centerofgravity motion (to be more precise, centerofenergy motion
in the relativistic case) and the relative motion. The "relative orbital
angular momentum" refers to the orbital angular momentum for the
relative coordinate x1x2. For the initial state of rho or omega
meson, the only angular momentum, after you separate out the
centerofenergy 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
I31/2,1/2>=(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!

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6421019
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