hw03sol - 1 Introduction to Quantum and Statistical...

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Unformatted text preview: 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 2 − = ∂ ∂ = h , where V ˆ is the same as in (a). (4 pts) (c) ] ˆ , ˆ [ 2 p p = 0 (4 pts) (d) 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 exp exp ˆ h = . Therefore the eigenfunction is the plane wave function with the corresponding eigenvalue of k h . Notice k 0 is a continuous parameter. (5 pts) (c) ( ) ( ) ( ) 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 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...
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This note was uploaded on 10/25/2009 for the course ECE 4060 at Cornell.

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hw03sol - 1 Introduction to Quantum and Statistical...

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