Lect17_WavePackScatt - Gaussian Wavepackets Review An exact...

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Lecture 17: Wavepacket Scattering and Classical limit I PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Gaussian Wavepackets Review: An exact solution to Schrodinger’s equation for a free- particle: For the initial condition: Probability density moves and spreads in time: Center moves as a classical particle would: Width has delay then linear growth in time: h h h ! " # $ % & + ! " # $ % & + ! " # $ % & ( ( ) * + + , - ! " # $ % & + = m t p x p i m t i x m t p x e m t i t x 2 1 2 4 / 1 2 0 0 2 2 2 0 0 ) , ( . / 0 ( ) ) ( ) ( 2 2 2 0 ) ( 1 ) , ( t t x x e t t x ! " # $ $ = t m p x t x 0 0 0 ) ( + = 2 2 1 ) ( ! " # $ % & + = m t t h ( ) h x p i x x e x 0 2 2 0 2 1 ) 0 , ( + ! ! = $ h 2 m t s = m v s h =
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Wavepacket Scattering Suppose a wave packet is incident upon a potential step: Initial state: We need to expand onto eigenstates of H: Basis: Defined via: 0 V ( ) x ik x x e x 0 2 2 0 2 1 ) 0 , ( + ! ! = " # $ E 0 0 < x 0 0 > k x k = 1 2 e ikx + r ( k ) e # ikx ( ) u ( # x ) + t ( k ) e iK ( k ) x u ( x ) [ ] { } k Normalization constant Unit step function Unit step function Basis Vectors Basis state for step potential are given by: Where: x k = 1 e ikx + r ( k ) e # ikx ( ) u ( # x ) + t ( k ) e iK ( k ) x u ( x ) [ ] ) ( ) ( ) ( k K k k K k k r + ! = t ( k ) = 2 K ( k ) k + K ( k ) E = h 2 k 2 2 m = V 0 + h 2 K k ( ) [ ] 2 2 m " K k ( ) = k 2 # 2 mV 0 h 2 u k # mV 0 ( ) + i u mV 0 # k ( ) [ ]
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Expansion: We need to write our initial state in terms of the basis states, and add the time evolution:
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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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Lect17_WavePackScatt - Gaussian Wavepackets Review An exact...

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