notes22 - 5.73 Lecture #22 22 - 1 3D-Central Force Problems...

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22 - 1 5.73 Lecture #22 revised 21 October, 2002 @ 10:19 AM 3D-Central Force Problems II Last time: [ x , p ] = i h vector commutation rules: generalize from 1-D to 3-D conjugate position and momentum components in Cartesian coordinates Correspondence Principle Recipe Cartesian and vector analysis Symmetrize (make it Hermitian) classical in h 0 limit Derived key results: fi xp f x rqp f r r x () [] = = ++ , , / h h based on d d and = f r r x rx y z 222 12 * * * *( ) pr q p ppr L r r Lqp H pL r r r =⋅ =+ = µ + µ + 1 22 2 2 2 2 2 i operator r V h algebra gave simple separation of variables not necessary (or possible) to symmetrize V radial effective potential l TODAY Obtain angular Momentum Commutation Rules Block diagonalize H ε ijk Levi-Civita Antisymmetric Tensor useful in derivations, vector commutators, and remembering stuff. Next Lecture: Begin derivation of all angular momentum matrix elements starting from Commutation Rule definitions of angular momentum. [purpose is mostly to practice [,] and angular momentum algebra] (came from symmetrization in Cartesian coordinates) We do not yet know anything about L 2 nd L i .
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22 - 2 5.73 Lecture #22 revised 21 October, 2002 @ 10:19 AM GOALS 10 20 30 4 00 5 2 2 2 ., ( ) ! .. . . . , , Lr Lp LL HL L H r r i i i i i f CSCO [] = = = = () any scalar function of scalar .
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notes22 - 5.73 Lecture #22 22 - 1 3D-Central Force Problems...

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