What are the values of j and m how to build jm

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Unformatted text preview: m J z jm = m jm ? J = J1 + J 2 Addition of the angular momentum states J 2 jm = j ( j + 1) jm J z jm = m jm How many states? • What are the values of j and m? • How to build jm ? Clebsch-Gordon coefficients ( j1 j2 ) JM = j1m1 j2 m2 m1, m 2 j1m1 j2 m2 JM L = l1 + l2 Example: two p electrons Example: l1=1 l1m1 l2=1 l2 m2 ' $ L=? M=? LM = ? ) * ( $' + $ " ! "# $ %& , - l " .- l " . ) + * L− LM = L( L + 1) − M ( M − 1) L, M − 1 Example: two p electrons Example: l1=1 l1m1 l2=1 l2 m2 * (l1l2 ) LM = l1m1 , l2 m2 (11)22 = 11,11 ' $ ) * + ( $' $ L− = l− (1) + l− (2) " L− (11)22 = 2 (11)21 = L− 11,11 = 2 10,11 + 2 11,10 (11)21 = 1 { 10,11 + 11,10 } 2 L− LM = L( L + 1) − M ( M − 1) L, M − 1 L+ LM = L( L + 1) − M ( M + 1) L, M + 1 Example: two p electrons Example: 0 ' $ 1 L− (11)21 = 6 (11)20 = L− { 10,11 + 11,10 } = 2 1 = 2 1 − 1,11 + 2 10,10 + 2 10,10 + 2 11,1 − 1 2 1 (11)20 = { 1 − 1,11 + 2 10,10 + 11,1 − 1 } 6 { 1 2 3 } ' " (11)2 − 2 = 1 − 1,1 − 1 L+ (11)2 − 2 = 2 (11)2 − 1 = L+ 1 − 1,1 − 1 (11)2 − 1 = 1 { 10,1 − 1 + 1 − 1,10 2 } Example: tw...
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