385lec7

# 385lec7 - PHYS 385 Lecture 7 Momentum operator Lecture 7 Momentum operator What's important momentum operator commutation relations Hermitian

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7 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 7 - Momentum operator What's important : momentum operator commutation relations Hermitian operators Text : Gasiorowicz, Chap. 3. Momentum operator Once we have solved the Schrödinger equation i h ( x , t ) t = - h 2 2 m 2 ( x , t ) x 2 (1) for some particular condition (we'll introduce the form of the SE which includes interactions in a moment) then we can calculate the mean position and its dispersion from quantities like < x > = * x dx and < x 2 > = * x 2 dx . (2) Given the form of the wavepackets described earlier, it's obvious how to do the mathematics of Eq. (1) - one just integrates over some functions which depend on x and t . But what about a quantity like the momentum p ? What is the meaning of the expression < p > = * p dx ? An appealing starting point is to return to the classical definition p = mv = m dx / dt and PROPOSE < p > = m d < x >/ dt . (3) We'll provide the motivation for this in a later lecture on the classical limit of . Thus, < p > = m ( d / dt ) * x dx . (4) The wavefunction is time-dependent, so taking the time derivative inside the integral will yield terms like d / dt . However, there is no time-dependence to x : it's just an integration variable, not a function of

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## This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.

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385lec7 - PHYS 385 Lecture 7 Momentum operator Lecture 7 Momentum operator What's important momentum operator commutation relations Hermitian

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