Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 : Lrp 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