Clebsch-Gordan Coefficents Addition of angular momenta J1 J2
Clebsch-Gordan Coefficents Addition of angular momenta J1 J2 1 j1m1 basis for J12 and J1z 2 j2 m2 basis for J 22 and J 2 z The base vectors j1 j2 m1 m2 1 j1 m1 2 j2 m2 , j1 , j2 fixed , m1 , m2 vary j1 m1 j1 j2 m2 j2 span the subspace ( , j1 , j2 ) . J 2 ( J1 J 2 ) 2 and J z act on ( , j1 , j2 ) Since J12 and J 22 commute with J 2 and J z , can also use the base j1 j2 j m , , j1 , j2 fixed
j , m vary j1 j2 j j1 j2 j m j to generate the same subspace ( , j1 , j2 ) . The two bases are related: j1 j2 j m j2 j1 j1 j2 m1m2 j1 j2 m1m2 jm m2 j2 m1 j1 j j1 j2 m1 m2 m j ( j1 j2 ) j1 j2 j m j m j1 j2 m1 m2 j j1 j1
j1 j2 m1 m2 j m j m j1 j2 m1 m2 * Clebsch-Gordan coefficents Meaning of C.G. coeffs (i) (ii) relating two basis vectors (just like Fourier transform) j1 j2 m1 m2 j m = probability amplitude of finding the state j1 j2 m1 m2 when the system is in state jm Properties of C.G. coeffs (1) Selection rule: j1 j2 m1m2 jm 0 unless m1 m2 m and j1 j2 j ( j1 j2 ) (2) Phase convention: require j1 j2 j1 m2 j j real and 0 m 2 j j1 j j1 j2 , j1 j2 1.... ( j1 j2 ) Note: When m1 j1 and m j , it does not necessarily imply m2 j2 since j ( j1 j2 ) in general (3) Reality: All C.G. coeffs can be obtained from j1 j2 j1 m2 j j
all C.G. coeffs are real (4) Orthogonality j1 j2 m1 m2 j m m1 m2 j m j1 j2 m1 m2 j m j1 j2 m1 m2 j ' m ' jj ' mm' j1 j2 m1' m2 ' j m m m ' m m ' 1 1 2 2 Wigner-Eckart theorem In a standard representation J 2 , J z whose basis vectors are denoted by jm , j m Tg( k ) ' j ' m' The matrix element of the qth standard component of a given kth order irreducible tensor
operator, T(k), is equal to the product of the Clebsch-Gordan coefficient. j ' k m 'q jm by a quantity independent of m, m and q j m Tq( k ) ' j 'm ' (q k , k 1,.... k ) 1 j | | T ( k ) || ' j ' j 'k m 'q jm 2 j 1 j | | T ( k ) || ' j ' = reduced matrix element j ' k m' q jm =Clebsch-Gordan coefficient 0 only if m m ' q and For a scalar operator S j j ' k j j ' jm s ' j ' m ' jj ' mm' S( j' ) Summary e e Wave functions ( x ) e
ipx / u ( s ) ( p) ( x ) e ipx / ( s ) v ( p) s=1 spin up s=1 spin down s=1 spin down s=2 spin up ( p mc)u 0 ( p mc)v 0 u ( p mc) 0 v ( p mc ) 0 Orthonormality u ( s1 )u ( s2 ) 2mc s1s2 v ( s1 ) v ( s2 ) 2mc s1s2 s1, s2 = 1, 2 Completeness 2
2 (s) u u s 1 (s) ( p mc ) (s) (s) v v ( p mc) s 1 Summary Photon Plane Wave A ( x ) e ipx / (s ) , s=1, 2 for the two polarization states Polarization vector statistics, p 0 Orthonormality * s1 ( s2 ) s1s2
Coulomb gauge 0, p 0 Completeness 2 ( s 1 ) ( (*s ) ) j ij pi p j (s) i p i pi / | p | Feynman rules QED p 4 s4 p5 s5 p6 s6 (1) Notations Label external lines by momentum pi and spin si , Label internal lines by momenta qi
p1s1 p2 s2 p3 s3 Arrows on external fermion lines indicate e (forward in time) e (backward in time) Arrows on internal fermion lines are assigned so that direction of the flow of 4-momenta through the diagram is kept. Arrows on external photon lines point forward; for internal photon lines, the choice is arbitrary. (2) External lines e (3) incoming outgoing :u :u e incoming
outgoing :v :v incoming outgoing : Vertex Each vertex contributes a factor ig g= dimensionless coupling constant = 4 : * (4) Propagators (internal lines) i ( q mc) i e or e : q mc q 2 m 2 c 2 :
ig v q2 (5) Conservation of 4 - momentum P : k2 k3 k1 For each vertex, write (2 ) 4 (4) ( k1 k2 k3 ) (6) Integrate over internal momenta d 4q (2 )4 (7) Cancel the overall delta function (2 ) 4 (4) ( p1 p2 .... pn ) what remains is the -i = scattering amplitude (8) Include a minus sign between diagrams that differ only in the interchange of two incoming (or outgoing) e ' s (or e ' s ) or of an incoming e with an outgoing e (or vice versa) (9) Charge is conserved at each vertex. Lepton number etc must also be conserved.
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