Lect13_ClassicalLimit_pre

# Lect13_ClassicalLimit_pre - Lecture 13 The classical limit...

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Lecture 13: The classical limit Phy851/fall 2009 ?

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Wavepacket Evolution For a wavepacket in free space, we have already seen that – So that the center of the wavepacket obeys Newton’s Second Law (with no force): Assuming that: – The wavepacket is very narrow – spreading is negligible on the relevant time- scale Would Classical Mechanics provide a quantitatively accurate description of the wavepacket evolution? – How narrow is narrow enough? What happens when we add a potential, V ( x ) ? – Will we find that the wavepacket obeys Newton’s Second Law of Motion? 0 0 0 p p t M p x x = + = 0 = = p dt d m p x dt d
Equation of motion for expectation value How do we find equations of motion for expectation values of observables? – Consider a system described by an arbitrary Hamiltonian, H – Let A be an observable for the system – Question: what is: Answer: ? A dt d ) ( ) ( t A t dt d A dt d ψ = + + = ) ( ) ( ) ( ) ( ) ( ) ( t dt d A t t t A t t A t dt d d dt A = i h A , H [ ] + A t t A t AH t i t HA t i + = ) ( ) ( ) ( ) ( h h

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Example 1: A free particle Assuming that The basic equation of motion is: For a free particle, we have: M P H 2 2 = X , H [ ] = 1 2 M X , P 2 [ ] d dt A = i h A , H [ ] = 1 2 M XP 2 P 2 X ( ) = 1 2 M XP 2 PXP + PXP P 2 X ( ) = 1 2 M X , P [ ] P + P X , P [ ] ( ) M P i h = H , P [ ] = 0 m
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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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Lect13_ClassicalLimit_pre - Lecture 13 The classical limit...

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