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Lect23_AngMomentI

# Lect23_AngMomentI - Goals 1 Derive the quantum mechanical...

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Lecture 23: Angular Momentum I PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Goals 1. Derive the quantum mechanical properties of Angular Momentum 2. Use an algebraic approach similar to what we did for the Harmonic Oscillator 3. Use the resulting theory to treat spherically symmetric problems in three dimensions

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Motion In 3Dimensions For a particle moving in three dimensions, there is a distinct quantum state for every point in space. Thus each position state is now labeled by a vector Vector operators are really three operators Coordinate system not unique, e.g.: R X v ! r x r ! z y x e Z e Y e X R r r r r + + = Z Y X , , Scalar Operators Ordinary Vectors z y x e e e r r r , , ) , ( ! " = r e R R r r ! " , , R Scalar Operators z y x r e e e e r r r r ! + " ! + " ! = " ! cos sin sin cos sin ) , ( Vector Operator ) , ( ! " r e r You can never go wrong with Cartesian Coordinates In all other coordinate systems, the unit vectors are also operators so must be treated carefully Eigenstates: For each point in space there is a position eigenstate
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Lect23_AngMomentI - Goals 1 Derive the quantum mechanical...

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