Lect6_SchrodEq - Review of X and P representations A...

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Review of X and P representations A particle has two observables: Position, X , and momentum, P Each has a continuous spectrum: Commutation relation: From commutation relation we can derive: Action of P : Momentum eigenstate wavefunction: [ ] h i P X = , ) ( x x x x ! " = ! # x x x X = 1 = ! x x dx ) ( p p p p ! " = ! p p p P = 1 = ! p p dp ) ( x x dx x x dx ! " " = = ) ( p p dp p p dp " " = = ) ( ) ( ) ( x x i x x x x i x P x ! " ! = ! " ! " = ! h h h h / 2 1 ipx e p x = x x i P x " " # = h Lecture 6: Solving Schrödinger's Equation General Form: Specific to motion of a particle: Note about V ( X ) : V ( X ) is an operator. Eigenstates of V ( X ) are the position eigenstates, | x ! . In general, if: then: Proof: x x V x m x dt d i " # $ % + ( ( ) = ) ( 2 2 2 2 h h H dt d i = h ) ( 2 2 X V m P H + = a a a A = a a f a A f ) ( ) ( = a a f a a f a A f a A f m m m m m m ) ( ) ( = = = ! ! Recall that a function is defined by its power-series
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Methods of Solving Schrödinger’s Equation When the Hamiltonian is not explicitly time- dependent, Schrödinger's Equation is readily integrated: Proof: ) ( ) ( t
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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Lect6_SchrodEq - Review of X and P representations A...

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