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Chapter4_12 - Also note that(J op(J op =(J 2 op(J z op(J z...

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PHY4604 R. D. Field Department of Physics Chapter4_12.doc University of Florida Lie Algebra – Generalized Angular Momentum Lie Algebra: Consider a finite number of operators X i that satisfy the commutation rules: = k op k ijk op j op i X C X X ) ( ] ) ( , ) [( If commuting any two of them is a linear combination of the other operators in the set then the operators are said to form a “Lie Algebra” where the constants C ijk are called the “structure constants” of the algebra. From these operators one can construct operators Y i called “Casimir” operators that commute with all the operators in the set: 0 ] ) ( , ) [( = op j op i X Y for all i and j. Generalized Angular Momentum Operators: Let h / 1 x x L J J = = , h / 2 y y L J J = = , and h / 3 z z L J J = = . Then op i op i J J ) ( ) ( = and op k ijk op j op i J i J J ) ( ] ) ( , ) [( ε = . The operators J i form a “Lie Algegra” with “structure constants” ε ijk . The operator (J 2 ) op = (J x ) 2 op +(J y ) 2 op + (J z ) 2 op is a “Casimir” operator since 0 ] ) ( , ) [( 2 = op i op J J for i = 1, 2, 3. Raising and Lowering Operators: Define op y op x op J i J J ) ( ) ( ) ( ± = ± and then 0 ] ) ( , ) [( 2 = ± op op J J and op op op z J J J ) ( ] ) ( , ) [( ± ± ±
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Unformatted text preview: . Also note that (J + ) op (J-) op = (J 2 ) op- (J z ) op ((J z ) op-1) and (J-) op (J + ) op = (J 2 ) op- (J z ) op ((J z ) op +1). Eigenkets of (J 2 ) op and (J z ) op : Define |jm> to be an eigenket of (J 2 ) op and (J z ) op with eigenvalues given by (J 2 ) op |jm> = j(j+1)|jm> and (J z ) op |jm> = m|jm> with <j'm'|jm> = δ j'j δ m'm . We see that (J 2 ) op ((J ± ) op |jm>)= (J ± ) op (J 2 ) op |jm> = j(j+1) ((J ± ) op |jm>) (J z ) op ((J ± ) op |jm>)=( (J ± ) op (J z ) op ±(J z ) op )|jm> = (m±1) ((J ± ) op |jm>) and hence (J ± ) op |jm> = c ± |jm±1> . The operators (J ± ) op shifts the state |jm> to a new state |jm±1> with the same j value but with m shifted by ±1 . Note that at this point we do not know anything about j and m except that they are real numbers ( we are not assuming that they are integers ). Orthonormal set of states!...
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