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Lect26_CentralPotential - Review Orbital Angular Momentum...

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Lecture 26: Motion in a Central Potential PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Review: Orbital Angular Momentum ! " " # $ h i L z ! ! " # $ $ % & + ( ) 2 2 2 2 2 sin 1 sin sin 1 * + + + + + h L m m m L m m L z , , , ) 1 ( , 2 2 l h l l l l h l = + = K l , 3 , 2 , 1 , 0 = l K l l , , 1 , + ! ! = m ) , ( , , ! " ! " m Y m l l = 2 2 2 mr L T T r + = ! ! " # $ $ % & + ( = r r r m T r 2 2 2 2 2 h
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Motion in a Spherically Symmetric Potential Now lets consider the special case of a potential that depends only on the radius r. Let: The total Hamiltonian is then: Both L 2 and L z commute with H : This means we can find simultaneous eigenstates of H , L 2 and L z : New basis set (assuming a discrete spectrum): Defined by: ) ( ) ( r V r V = r 0 ] , [ 2 = R L ) ( 2 2 2 R V mR L T H r + + = n , l , m { } 0 ] , [ 2 = r T L 0 ] , [ = r z T L m n E m n H n , , , , l l = m n m n L , , ) 1 ( , , 2 2 l l l h l + = m n m m n L z , , , , l h l = 0 ] , [ = R L z Derivation of Radial Wave Equation We start from the energy eigenvalue equation: Hit with | r l m ! eigenstate from left: Define the radial wavefunction: To get: Note: m n E m n H n , , , , l l = m n m r E m n H m r n , , , , , , , , l l l l
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