# hw03sol - Introduction to Quantum and Statistical Mechanics...

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Introduction to Quantum and Statistical Mechanics Homework 3 Solution Prob. 3.1. (a) x ˆ and 2 ˆ x commutate, so )] ( ˆ , ˆ [ x V x = 0 (4 pts) (b) p x ω m x x ω m i x V p ˆ ˆ )] ( ˆ , ˆ [ 2 0 2 0 - = = , where V ˆ is the same as in (a). (4 pts) (c) ] ˆ , ˆ [ 2 p p = 0 (4 pts) (d) 0 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ] ˆ ˆ , ˆ ˆ [ 2 2 2 2 2 2 2 2 2 2 2 2 2 = + - + = + - = - = x x x x x x x x x x x x x x p x p x p x x p p x V V V (4 pts) Prob. 3.2 For the momentum operator p ˆ , (a) k x i p V V = - = ˆ (4 pts) (b) ( 29 ( 29 x ik k x ik p 0 0 0 exp exp ˆ ˆ = . Therefore the eigenfunction is the plane wave function with the corresponding eigenvalue of 0 k V . Notice k 0 is a continuous parameter. (5 pts) (c) ( 29 ( 29 ( 29 0 0 0 0 ˆ k k k k k k k k p - = - = - δ . Therefore the eigenfunction is the delta function with the corresponding eigenvalue of

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## This note was uploaded on 11/07/2009 for the course ECE 4060 at Cornell University (Engineering School).

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hw03sol - Introduction to Quantum and Statistical Mechanics...

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