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10 lectures
2
classical physics:a physical system is given by the functions of the coordinates and
of the associated momenta –
)(),( tptq ii
3
quantum physics:
coordinates and momentaare Hermitean operators in the
Hilbert space of states
hixi
hxpx
,,
4
*:yprobabilit
),...,,,,,(: 321321 tyyyxxxfunctionwave
5
1927:Schrödingerequation
for thewave
function
6
7
2
2
2
2
2
2
2
2
),(
zyx
Vm
hH
trHt
hi
8
9
harmonic oscillator
11
12
13
14Gauß curve
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16
Symmetry in
Quantum Physics
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•A. external symmetries
B. internal symmetries
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external symmetries:Poincare group
conservation laws:
energy momentum angular momentum
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external symmetries:
exact in Minkowski space
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General Relativity:no energy conservation
no momentum conservationno conservation of angular
momentum
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internal symmetries:
IsospinSU(3)
Color symmetryElectroweak gauge symmetry
Grand Unification: SO(10)Supersymmetry
22
internal symmetries
broken by interaction: isospin
broken by quark masses: SU(3)
broken by SSB: electroweak symmetry
unbroken: color symmetry
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symmetriesare described
mathematicallyby groups
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symmetrygroups
nassociatiooflawgggggg
productofdefinitionggg
ggggG
cbacba
dba
n
)...,,(: 321 n finite or infinite
eggggelementinverse
geggeidentity
aaaa
aaa
11:
:
25
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
26
numbersrealge
groupAbeleananisG
elementsgroupallforggggif
groupAbeleanNon
gggggeneralin
abba
abba
..
:
:
27
Niels AbelNorway
19th century
28
A symmetry is a transformation of
the dynamical variables,
which leave the action invariant.
29
Classical mechanics:
translations ofspace and time –
( energy, momentum )rotations of space
( angular momentum )
30
Special Relativity =>Poincare group:translations of
4 space - time coordinates+
Lorentz transformations
31
Henri Poincare
late19th century
32
Symmetry in quantum physics ( E. Wigner, 1930 … )
U
U: unitary operator
33
Eugene Wigner
Nobel prize1964
34
PaiaU
txaUtax
ntranslatiospace
exp)(
),()(),(
.1
35
ibHbU
txbUbtx
ntranslatiotime
exp)(
),()(),(
.2
36
z
kijkji
z
y
x
iL
axiszaroundrotation
LiLL
L
L
L
LU
txRUtxR
rotationspace
exp
:
,
),()(),(
.3
37
Poincare group P: - time translations -- space translations -- rotations of space -- „rotation“ between
time and space -
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e.g. rotations of space:
kijkji LiLL ,-
-
SO(3)< P
39
Casimir operatorof Poincare group
operatormass
MPP 2
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0, UH
The operator U commutes with the Hamiltonoperator H:
If U acts on a wave function with a specific energy, the new wave function must have the same energy ( degenerate energy levels ).
symmetry U
41
statesL
tripletl
ln
energysamethewithstatesofnumber
momentumangularexample
3
1
0
1
:
1
12
:
:
3
42
discrete symmetries
P ~ parity symmetry
CP - symmetry
CPT - symmetry
43
PLL
prLmomentumangular
ppmomentum
xx
:
:
44
)()(
1
1
:
1
)()(2
xx
parityoddp
parityevenp
Pofeigenvaluep
P
xxP
45
P: exact symmetry in the strong and electromagnetic
interactions
46
P: maximal violation
in theweak interactions
47
momentumtooppositespine
lefthandedelectron
epn
decay
e
:
:
:
48
theory of parity violation:1956:
T. D. Lee and C.N.Yang
experiment: Chien-Shiung Wu
( Columbia university )
49
Lee Yang Wu
50
Experiment of Wu:beta decay of cobalt
eeNiCo 6028
6027
51
Co Co
R
L
52
electrons emitted primarily against Cobalt spin
( violation of parity )
Co Co
R
L
53
1958
Feynman, Gell-MannMarshak, Sudarshan
maximal parity violationlefhanded weak currents
54
CP – violation:
weak interactions wereCP 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 )
55
present theory of CP-violation:
phase in the mixing matrixof the quarks
56
Klein-Gordon equation
Oscar Klein and Walter Gordon Stockhom, 1927
relativistic versionof Schrödinger equation
222 mpE
c = 1
Klein
57
58
59
V. Weisskopf – W. Pauli (~1933)
the Klein-Gordon field is not a wave function, but describes a scalar field
60
61
62
63
64
Spin
Dirac equation
65
Experiment ofStern and Gerlach
(1922)
66
67
The beam of silver atoms
was split intotwo beams
68
Goudsmit – Uhlenbeck1924
a new discretequantum number
69
70
Spin1927
2
1,2
1
2
1,2
1
h2
1
spin
71
01
101
0
02 i
i
10
013
Pauli matrices
72
angular momentum:
kijkki LiLL ,
73
kijkijji sissss
2
1s
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 )
Dirac equation
79
4
3
2
1
80
0)( mi
4
3
2
1
81
4
3
2
1
lesantipartic
82
83
positron
84