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notes23 - 5.73 Lecture#23 Angular Momentum Matrix Elements...

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23 - 1 5.73 Lecture #23 revised 4 November, 2002 Angular Momentum Matrix Elements LAST TIME: * all [, ]=0 Commutation Rules needed to block diagonalize * ε ijk Levi-Civita antisymmetric tensor — useful properties * Commutation Rule DEFINITIONS of Angular Momentum and “Vector” Operators TODAY: Obtain all angular momentum matrix elements from the commutation rule definition of an angular momentum, without ever looking at a differential operator or a wavefuncton. Possibilities for phase inconsistencies. [Similar derivation for angular parts of matrix elements of all spherical tensor operators, .] H p L r r = µ + µ + ( ) r V 2 2 2 2 2 in nLM basis set L L L L L V V i j ijk k k i j ijk k k i i , , [ ] = [ ] = ε ε 1. Define Components of Angular Momentum using a Commutation Rule. 2. Define eigenbasis for J 2 and J z | λ μ 3. show λ μ 2 4 raising and lowering operators (like a , a and x p ~ ~ ± i ) J ± | λ µ gives eigenfunction of J z belonging to µ ± eigenvalue and eigenfunction of J 2 belonging to λ eigenvalue 5. Must be at least one µ MAX and one µ MIN such that J ( J + | λ µ MAX ) = 0 J + ( J | λ µ MIN ) = 0 This leads to µ max = n n n 2 2 2 1 2 = + , λ . 6. Obtain all matrix elements of J x , J y , J ± , but there remains a phase ambiguity 7. Standard phase choice: “Condon and Shortley” T q k ( ) Classification of operators: universality of matrix elements.
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23 - 2 5.73 Lecture #23 revised 4 November, 2002 1. Commutation Rule This is a general definition of angular momentum (call it J , L , S , anything!) . Each angular momentum generates a state space.
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