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```10 lectures
classical physics:
a physical system is given by the
functions of the coordinates and
of the associated momenta –
qi (t ),
pi (t )
2
quantum physics:
coordinates and momenta
are Hermitean operators in the
Hilbert space of states
 h  
x, p    x,

i
h

 i x 
3
wave
function :  ( x1 , x2 , x3  y1 , y2 , y3 ,..., t )
probabilit y :  
*
4
5
6


ih
 H ( r , t )
t
2
h
H 
 V
2m
2
2
2






2
2
2
x
y
z
7
8
9
11
12
13
Gauß curve
14
15
Symmetry
in
Quantum Physics
16
•A. external symmetries
B. internal symmetries
17
external symmetries:
Poincare group
conservation laws:
energy
momentum
angular momentum
18
external
symmetries:
exact in
Minkowski space
19
General Relativity:
no energy conservation
no momentum conservation
no conservation of angular
momentum
20
internal symmetries:
Isospin
SU(3)
Color symmetry
Electroweak gauge symmetry
Grand Unification: SO(10)
Supersymmetry
21
internal symmetries
broken by interaction: isospin
broken by quark masses: SU(3)
broken by SSB: electroweak symmetry
unbroken: color symmetry
22
symmetries
are described
mathematically
by groups
23
symmetry
groups
G : ( g1 , g 2 , g 3 ...g n )
g a  gb  g d
n finite or infinite
definition of
g a  g b g c    g a g b g c
law of
product
associatio n
identity : e  g a  g a  e  g a
1
a
1
a
a
inverse element : g a  g  g g  e
24
examples of groups:
integer numbers:
3 + 5=8, 3 + 0=3,
5 + (-5) = 0
real numbers:
3.20 x 2.70=8.64, 3.20 x 1 = 3.20
3.20 x 0.3125 = 1
25
in
general : g a g b 
 Non  Abelean
if
g a gb  gb g a
gb g a
group
for all
 G is
an
e.g . real
numbers
Abelean
group elements :
group
26
27
A symmetry is a
transformation of
the dynamical
variables,
which leave the
action invariant.
28
Classical mechanics:

translations of
space and time –
( energy, momentum )
rotations of space
( angular momentum )
29
Special Relativity =>
Poincare group:
translations of
4 space - time coordinates
+
Lorentz transformations
30
31
Symmetry in quantum physics
( E. Wigner, 1930 … )
 U 
U: unitary operator
32
33
1. space translatio n
 


 ( x  a , t )  U (a )  ( x , t )


U (a )  exp iaP
34
2. time translation


 ( x , t  b)  U (b)  ( x , t )
U (b)  exp  ibH
35
3. space rotation


 ( Rx , t )  U ( R)  ( x , t )
U
 Lx 
  
L   Ly 
L 
 z
rotation around
exp iL z
L , L   i
i
j
ijk
Lk
z  axis :
36
Poincare group P:
- time translations - space translations - rotations of space - „rotation“ between
time and space 37
e.g. rotations of space:
L , L   i
-
-
i
j
L
ijk k
38
Casimir operator
of Poincare group

P P  M
2
mass operator
39
The operator U commutes
with the Hamiltonoperator H:
H ,U   0
If U acts on a wave function with a specific energy,
the new wave function must have the same energy
( degenerate energy levels ).
40
example : angular
number of
n  2l  1
l 1
triplet
L3 :
  1
 
0
  1
 
momentum
states with the same energy :
3 states
41
discrete symmetries
42


x  x


momentum : p   p
angular momentum :
  
Lrp


L  L
P
43


P  ( x )   ( x )
P 1
2
p : eigenvalue of
P
p  1
even
parity
p  1
odd
parity


 ( x )    ( x )
44
P: exact symmetry
in the strong and
electromagnetic
interactions
45
P:
maximal violation
in the
weak interactions
46
  decay :
n  p  e  e

electron : lefthanded

e :


spin opposite to momentum
47
theory of parity violation:
1956:
T. D. Lee and C.N.Yang
experiment:
Chien-Shiung Wu
( Columbia university )
48
Lee Yang Wu
49
Experiment of Wu:
beta decay of cobalt
Co Ni  e  e
60
27
60
28

50
51
electrons emitted primarily
against Cobalt spin
( violation of parity )
52
1958
Feynman, Gell-Mann
Marshak, Sudarshan
maximal parity violation
lefhanded weak currents
53
weak interactions were
CP invariant, until
1964:
CP violation found at the level of
0.1% of the parity violation
in decay of neutral K-mesons
(James Cronin and Val Fitch, 1964 )
54
present theory of
CP-violation:
phase in the mixing matrix
of the quarks
55

2
2
2
E p m
56
57
58
V. Weisskopf – W. Pauli
(~1933)
the Klein-Gordon field is
not a wave function,
but describes a scalar field
59
60
61
62
63
64
65
66
67
68
Goudsmit – Uhlenbeck
1924
a new discrete
quantum number
69
70
1
1
,

2
2
1
h
2
1
1
,

2
2
71
0 1

 1  
1 0
0  i

 2  
i 0 
1 0 

 3  
 0  1
72
angular momentum:
Li , Lk   i ijk Lk
73

1 
s 

2
si s j  s j si  i ijk s k
74
Spin of particles:
pi-meson: 0
electron, proton: ½
photon: 1
delta resonance: 3/2
graviton: 2
75
76
matter particles have spin ½
=> fermions
( electron, proton, neutron )
force particles have spin 1
=> bosons
( photon, gluons, weak bosons )
77
78
Klein-Gordon equation:
no positive definite probability density exists
Dirac 1927:
search for a wave equation, in which
the time derivative appears only in
the first order
( Klein- Gordon equation:
second time derivate is needed )
79
 1 
 
2


 3 
 
 4 
80
(i    m)  0

 1 
 
 2 
  
3
 
 
 4
81
 1 
 
2


 3 
 
 4 
 antiparticles
82
83
positron
84
```
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