Problems Week 7
1. Consider the sum of two light waves
ψ = cos(( ¯K − ¯L) · R) + cos(( ¯K + ¯L) · R) ,
with ¯K, ¯L linearly independent. Write this as a wave A cos( ¯K · ¯R) with
modulated amplitude. Show that
¯K2
+ ¯L2
= ¯K · ¯L = 0 .
Show also that one of ¯K, ¯L must be timelike and the other spacelike.
Let ¯K be timelike and pick an observer with four-velocity ˆK. What
does the wave look like as measured by him?
2. Show that a photon (massless particle) cannot decay into an electron
and a positron (both with mass m > 0). Show that the reverse process
is also impossible. Show also that an electron cannot go into an electron
and a photon.
3. A particle of mass M decays into two particles with masses m1, m2.
What are the energies of the three particles as measured by an observer
at rest with respect to the decaying particle?
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