Neutrino Physics
Autumn semester 2013
Exercise 2
Drop into the box in the lobby by Wed 2.10. at 16.00.
1) Write the Dirac equation of a free fermion in terms of quantum mechanical energy
operator Ê = ( Hamiltonian) and three-momentum operator p̂ and show that the
Hamiltonian is given by H = g 0g × p̂ + mg 0 . Show then that the helicity h =
constant of motion in the case of free Dirac particle, that is, [ H,h ] = 0 .
S × p̂
is a
p
2) Show the following results concerning the spinors u(p,h) and v( p,h) :
u( p,h)g m u( p,h´) = v ( p,h)g m v( p,h´) = 2 p md hh´ ,
u( p,h)g 5u( p,h´) = v ( p,h)g 5 v( p,h´) = 0,
u † ( p,h)v( pP ,h´) = v† ( p,h)u( pP ,h´) = 0,
where pP = ( p0 ,- p). (Consult Giunti&Kim, p 24, link in the University ebrary is The
Fundamentals of Neutrino Physics and Astrophysics.)
3) The helicity of the neutrino was measured for the first time in an experiment carried
out by Goldhaber et al. in 1958. In this experiment the following electron capture (the
nucleus absorbs an orbital electron, resulting in the transformation of a proton into a
neutron in the nucleus and the emission of a neutrino) process was considered:
152
Eu + e- ® n e +152 Sm*®152 Sm + g ,
where the decay happened at rest. The helicity of the neutrino was determined by
measuring the helicity of the photon. The mother nucleus has spin = 0 and the
daughter nucleus 152 Sm * has spin =1. The spin of the photon is 1.
a) The photon is emitted in this reaction to the direction of the momentum of the
recoil nucleus 152 Sm . Argue why pg = - pn e .
b) Use the conservation of the angular momentum to show that the photon and the
neutrino should have their spins in the opposite directions.
c) How was the spin direction of the photon measured in this experiment? (Consult
the literature.)