Chapter4_14 - PHY4604 R D Field Addition of Angular...

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PHY4604 R. D. Field Department of Physics Chapter4_14.doc University of Florida Addition of Angular Momentum (1) Two Commutating Vectors: Suppose we have two vector operators op J ) ( 1 r and op J ) ( 2 r with 0 ] ) ( , ) [( 2 1 = op op J J r r and each of the vectors obey the same “lie algebra” op k ijk op j op i J i J J ) ( ] ) ( , ) [( 1 1 1 ε = and op k ijk op j op i J i J J ) ( ] ) ( , ) [( 2 2 2 ε = . The states |j 1 m 1 > are the eigenkets of op J ) ( 2 1 and op z J ) ( 1 and the states |j 2 m 2 > are the eigenkets of op J ) ( 2 2 and op z J ) ( 2 as follows: > >= > + >= 1 1 1 1 1 1 1 1 1 1 1 1 2 1 | | ) ( | ) 1 ( | ) ( m j m m j J m j j j m j J op z op > >= > + >= 2 2 2 2 2 2 2 2 2 2 2 2 2 2 | | ) ( | ) 1 ( | ) ( m j m m j J m j j j m j J op z op Form the Vector Sum: Let op op op J J J ) ( ) ( ) ( 2 1 r r r + = or op i op i op i J J J ) ( ) ( ) ( 2 1 + = for i = 1,2, 3. The “total angular momentum” op J ) ( r obeys the same “lie algebra” as op J ) ( 1 r and op J ) ( 2 r . Proof: op k ijk op k ijk op k ijk op j op i op j op i op i op j op i op i op j op i J i J i J i J J J J J J J J J J ) ( ) ( ) ( ] ) ( , ) [( ] ) ( , ) [( ] ) ( ) ( , ) ( ) [( ] ) ( , ) [( 2 1 2 2 1 1 2 1 2 1 ε ε ε = + = + = + + = We would like to construct the spectrum of states that are eigenkets of op J ) ( 2 and op z J ) ( . However, op op op op op J J J J J ) ( ) ( 2 ) ( ) ( ) ( 2 1 2 2 2 1 2 r r + + = and hence 0 ] ) ( , ) ( ) [( 2 ] ) ( , ) [( 1 2 1 1 2 = op z op op op z op J J J J J r r 0 ] ) ( , ) ( ) [( 2 ] ) ( , ) [( 2 2 1 2 2 = op z op op op z op J J J J J r r . Thus, we cannot specify simultaneously op z J ) ( 1 , op z J ) ( 2 , and op J ) ( 2 . However, 0 ] )
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