385lec4 - PHYS 385 Lecture 4 - Wavepackets Lecture 4 -...

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PHYS 385 Lecture 4 - Wavepackets 4 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 4 - Wavepackets What's important : wavepackets in position and momentum propagation of wavepackets Text : Gasiorowicz, Chap. 2 In the previous lecture, we derived the Fourier transform of a function f ( x ): f ( x ) = (2 π ) -1/2 - + A ( k ) exp( ikx ) dk . (1) where A ( k ) = (2 π ) -1/2 - + f ( x ) exp(- ikx ) dx . (2) Here, A ( k ) is a function of the continuous variable k , interpreted as a wavevector corresponding to the physical length x . Let's do a Fourier decomposition of a wavepacket, and then describe how the wavepacket propagates if k is treated as a momentum via p = h k . Wavepackets In Lec. 2, we describe how the uncertainty principle leads to the idea that motion in quantum mechanics is probabilistic - the position and momentum of a particle are described by probability densities P ( x ) and P ( p ) that have a distribution around some mean value: These functions look sort of Gaussian, as in exp(- k 2 ), where reflects the width of the distribution. Let's suppose that the WAVE AMPLITUDE of a particle in k -space is actually Gaussian, with a form g ( k ) = exp(- [ k - k o ] 2 ), (3) which is centered around k = k o . Compared to Eq. (2), we will drop the factors of (2 π ) 1/2 , as is done by Gasiorowicz; what interests us is the width of the distribution, not the normalization - in fact even with the missing (2 π ) 1/2 , Eq. (3) is not normalized to unity. How does the amplitude in position behave if the amplitude in momentum is Gaussian? P ( x vt x P ( p p o p
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PHYS 385 Lecture 4 - Wavepackets 4 - 2 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. We start by evaluating f ( x ) = - + exp(- [ k - k o ] 2 ) exp( ikx ) dk . (4) The first step in solving this integral is to combine the arguments of the exponentials - [ k - k o ] 2 + ikx then rewrite K k - k o - K 2 + i ( K + k o ) x . Completing the square gives
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385lec4 - PHYS 385 Lecture 4 - Wavepackets Lecture 4 -...

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