Lect14_Motion1D - Lecture 14: Motion in 1D Phy851/fall 2009...

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Lecture 14: Motion in 1D Phy851/fall 2009
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Simple Problems in 1D To Describe the motion of a particle in 1D, we need the following four QM elements: Putting them together yields the Schrödinger wave equation: ) ( ) ( t H t dt d i ψ = h ) ( 2 2 X V m P H + = ) , ( ) ( t x t x = ) , ( ) ( t x x i t P x = h ) , ( ) ( ) , ( 2 ) , ( 2 2 2 t x x V t x x m t x dt d i + = h h Schrödinger's equation Energy of a particle Definition of wavefunction Action of momentum operator in x-basis
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Bound States vs Scattering States Problems dealing with motion in 1D fall into one of two categories 1. Bound-state problems: V ( x ) < E over finite region only Energy levels are discrete Typical problem: Find Energy eigenvalues: { E n }; n =1,2,3,… Find corresponding Energy eigenstates: {| E n } Find time evolution of an arbitrary state 2. Scattering problems: V ( x ) < E in region extending to infinity in at least one direction Energy spectrum is continuous Typical problem: For a given incident k find reflection and transmission probabilities, R ( k ) and T ( k ) . V(x) E yes No
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E 0 Example: Scattering from a Step Potential Consider the potential: Goal: find eigenstates Strategy: – Divide into regions of constant V – Make suitable
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This note was uploaded on 12/21/2011 for the course PHY 851 taught by Professor Moore,m during the Fall '08 term at Michigan State University.

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Lect14_Motion1D - Lecture 14: Motion in 1D Phy851/fall 2009...

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