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Chapter7_8 - 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | | | | 10 | 2...

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PHY3063 R. D. Field Department of Physics Chapter7_8.doc University of Florida The Fundamental Representation of SU(2) The “fundamental” representation is the representation from which all other representations can be constructed. For the group SU(2) j = ½ (two component spinor) is the fundamental representation. Graphical Method: 2 × 2 = 3 + 1 m -1/2 2 of SU(2) +1/2 m -1 0 +1 m m 2 × 2 +1 0 -1 0 3 1 = + Operator Method: 2 × 2 = 3 + 1 Note that op op op y op x op J J J i J J ) ( ) ( ) ( ) ( ) ( 2 1 ± ± ± + = ± = Let j 1 = ½ and j 2 = ½ and apply the lowering operator to both sides of the highest state > > >= 2 1 2 1 2 1 2 1 | | 11 | , as follows: > > + >= 2 1 2 1 2 1 2 1 2 1 | | ) ) ( ) (( 11 | ) ( op op op J J J > > + + > > + >= + 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | | ) 1 ( ) 1 ( | | ) 1 ( ) 1 ( 10 | ) 1 1 ( 1 ) 1 1 ( 1 > > + > > >=
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Unformatted text preview: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | | | | 10 | 2 . Hence ( ) > − > + > > − >= 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | | | | 2 1 10 | where I used > − − − + >= − 1 | ) 1 ( ) 1 ( | ) ( jm m m j j jm J op . Applying the lowering operator to both sides of the |10> state gives > − > − >= − 2 1 2 1 2 1 2 1 | | 1 1 | Thus, ( ) > − > − >= − > − > + > > − >= > > >= 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | | 1 1 | | | | | 2 1 10 | | | 11 | ( ) > − > − > > − >= 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | | | | 2 1 00 | “3” of SU(2) “1” of SU(2)...
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