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

15

16

Symmetry in

Quantum Physics

17

•A. external symmetries

B. internal symmetries

18

external symmetries:Poincare group

conservation laws:

energy momentum angular momentum

19

external symmetries:

exact in Minkowski space

20

General Relativity:no energy conservation

no momentum conservationno conservation of angular

momentum

21

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

23

symmetriesare described

mathematicallyby groups

24

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 -

38

e.g. rotations of space:

kijkji LiLL ,-

-

SO(3)< P

39

Casimir operatorof Poincare group

operatormass

MPP 2

40

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