751 Take home test
December 16, 2009
1
1.1
Light Pulse
A pulse of light
y
y'
c
c
v
ct
x
ct'
vt
The invariance of the space-time interval is
s02 = s2
where
2
s2 = (ct) − x2 − y 2 − z 2
1
x'
2
s02 = (ct0 ) − x02 − y 02 − z 02
Due to the relative motion in our picture we really only have
2
2
(ct0 ) − x02 = (ct) − x2
where the primed and unprimed frames are related by
t−
t0 = p
x0 = px−ctv2
1−
v
c2
x
2
1− v2
c2
c
The two frames are therefore related by a Lorentz transformation like so
xµ = Lµν xν

β 0 0
 0 0 0
0
0
0
0
(x0 , x1 , x2 , x3 ) = 
 0 0 0
γ 0 0

x0
γ
 x1
0 

0   x2
β
x3




or something like that
1.2
Lorentz group parts
Time dependence
I really want to look this up
2
orthochronos
Rotations
Boosts
Spectra
2
Scleronomicity
Boosts
Rotations
m
0
-m
0
1
3
2
The Dirac Coulomb spectrum will not agree with any previous because it is
rst order in space and time derivatives
3
4
3
Spinless
E2
V0
E1
The classical electron E1 would reect back with no loss of momentum or
energy. The classical E2 would make it past the barrier but continue on with a
lower kinetic energy. The quantum nonrelativist says E1 reects a lower probability while the conserved probability penetrates the barrier with a decay(no
propagation). The quantum E2 non relativistic lepton will predict a tiny reection of prob. but a propagating solution for the particle past or within the
barrier.
4