# 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.

## Recently Viewed Presentations

• SLAMvs. STEM. Quantitative Literacy 1. Quantitative Literacy 2. Statistics. Liberal Arts Math. Beginning Algebra. Intermediate Algebra. STEM Track. Acceleration not a design goal, but there is a small amount.
• A Closer Look at the Suffix, -plasia -plasia means: condition of formation, development, growth Build medical terms with the following definitions by using -plasia. abnormal development excessive development incomplete development dysplasia hyperplasia hypoplasia Adjective Forms: dysplastic, hyperplastic, hypoplastic An Even...
• Fat. Specify healthful fats (low mercury/contaminant-containing nuts, avocado, certain plant oils, fish) Limit saturated fats (butter, fatty red meats, tropical plant oils, fast foods) and trans fat; choose fat-free or low-fat dairy products . Protein
• Surdas, Tulsidas, Meerabai, Nanak, Kabir, ChaitanyaMahaprabhu and Tyagaraja are few of the notable composers. Saint Valmiki. Ramayana the great epic of Hinduism is written by Saint Valmiki. Story of Saint Valmiki. Valmikirishi was previously known as "Valya", the robber. In...
• 2013 AEE / ASHRAE / USGBC. Energy Expo. October 2, 2013. Energy-Efficient Operation ManualsWhy do we need them?What does one looks like? Goal of Energy-Efficient Operation. To ensure that each significant energy-consuming device uses only as much energy as necessary...
• Horatio Alger Myth/"Rags to Riches" Novelist, Horatio Alger, wrote books about young men in rags to riches situations. Gave false hopes to many. Based stories off of Andrew Carnegie story. Most wealthy businesspeople of the day were: white, anglo, protestant...
• New Jersey offers great examples of the difference between local and regional perceptions of "place" compared the Census Bureau's definition of "place." Are townships "places?" According to New Jersey municipal code, the answer is yes. According to the Census Bureau's...
• What the law says about education for young people in the UKThe age young people 'leave school' in the UK is 18 years, although in practice most young people continue until the end of the academic year in which they...