Lecture_12

# Lecture_12 - PHYS PHYS342 Fall2011 Lecture 12 Fourier...

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HYS 342 PHYS 342 Fall 2011 ecture 12: Fourier Transforms and avepackets Lecture 12: Fourier Transforms and Wavepackets Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 12

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What we will show If a particle (represented as a wave) has an uncertainty in position, then there is an uncertainty in e momentum of that particle also You cannot the momentum of that particle also. You cannot measure both momentum and position of a quantum particle with high precision at the same time! width of wave uncertainty in position x
Wave Packets For the barrier transmission problems we have already For th arr r transm ss on pro ms w ha a r a y discussed, the wave function Ψ had a constant amplitude A and extended from x = - to x = + . These electron states are completely delocalized. mp y . ikx Ae  Q: How can we write a Ψ wavefunction that is more localized in space, to better approximate the motion of a particle? A: Use the Fourier transform x  1 () 2 ikx xk e dk    1 2 ikx kx e d x   where

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What the Equations Say () x Real space x (m)  1 2 ikx x ke dk   1 ikx kx e d x   2  ) o r g k k (m -1 ) ko k-space, or wavevector space, or reciprocal space
from a Fourier Series to the Fourier Integral http://phet.colorado.edu/simulations/sims.php?sim=Fourier_Making_Waves

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A Fundamental Property of Fourier Transform Pairs 3/ k rad mm / rad mm 0.106 x mm d  0.317 k x mm d 1 kx rad  1 rad 

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Lecture_12 - PHYS PHYS342 Fall2011 Lecture 12 Fourier...

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