Chapter4_24 - >=>= 1 | 10 | z − => −>>=...

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PHY4604 R. D. Field Department of Physics Chapter4_24.doc University of Florida Rotations and Angular Momentum Let I i J v h r = and hence = 0 1 0 1 0 0 0 0 0 h i J x = 0 0 1 0 0 0 1 0 0 h i J y = 0 0 0 0 0 1 0 1 0 h i J z The matrix operators J x , J y , and J z are hermitian and traceless and form a “lie algebra” with op k ijk op j op i J i J J ) ( ] ) ( , ) [( ε h = and 0 ] ) ( , [ 2 = op i op J J where = + + = 1 0 0 0 1 0 0 0 1 2 2 2 2 2 2 h z y x op J J J J and > + >= = = >= v j j v v v v v v v v J op | ) 1 ( | 2 2 1 0 0 0 1 0 0 0 1 2 | 2 2 3 2 1 2 3 2 1 2 2 h h h h Vectors are eigenkets of (J 2 ) op with eigenvalue j = 1 ! Eigenkets of (J z ) op : We must solve the eigenvalue equation > >= v m v J op z | | ) ( h = − = 3 2 1 1 2 3 2 1 0 0 0 0 0 0 1 0 1 0 v v v m v v i v v v i h h h or 0 3 2 1 1 2 = mv mv iv mv iv The three eigenvalues are m=1,0, -1 and the three normalized eigenkets are ( ) = > + > >= 0 1 2 1 | | 2 1 11 | i y i x
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Unformatted text preview: >= >= 1 | 10 | z ( ) − = > − > >= − 1 | | 2 1 1 1 | i y i x where the “unit” vectors are >= 1 | x >= 1 | y >= 1 | z . Proof: ± ± = ± − 1 1 1 1 i i i h h 1 1 1 = − h i...
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