hw03sol - Introduction to Quantum and Statistical Mechanics...

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1 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 = = h , 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 h h h (4 pts) Prob. 3.2 For the momentum operator p ˆ , (a) k x i p h h = = ˆ (4 pts) (b) ( ) ( ) x ik k x ik p 0 0 0 exp exp ˆ h = . Therefore the eigenfunction is the plane wave function with the corresponding eigenvalue of 0 k h . Notice k 0 is a continuous parameter. (5 pts) (c) ( ) ( ) ( ) 0 0 0 0 ˆ k k k k k k k k p = = δ δ δ h h . Therefore the eigenfunction is the delta function with the corresponding eigenvalue of 0 k h . Notice k 0 is a continuous parameter. (5 pts) (d) The Fourier transform of the plane wave function in the x space is the delta function in k space. Both the plane wave function and the delta function with continuous k0 can form a complete set of functions to express arbitrary nonpathological functions.
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