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**Unformatted text preview: **1 Introduction to Quantum and Statistical Mechanics Homework 3 Prob. 3.1. Find the following commutators. Remember that when the commutator of two operators does not vanish, it implies that the two operators cannot be determined with uncertainty simultaneously (a generalization of the Uncertainty Principle) (a) )] ( ˆ , ˆ [ x V x , where 2 2 ˆ 2 1 ˆ x m V ω = with m being the mass and ω the angular frequency later introduced in harmonic oscillators. (4 pts) (b) )] ( ˆ , ˆ [ x V p , where V ˆ is the same as in (a). (4 pts) (c) ] ˆ , ˆ [ 2 p p (4 pts) (d) ] ˆ , ˆ ˆ [ x p p x (4 pts) Prob. 3.2 For the momentum operator p ˆ , (a) Write down the explicit form of p ˆ in the x and k spaces. (4 pts) (b) What are the eigenvalues and eigenfunction in the x space? (5 pts) (c) What are the eigenvalues and eigenfunctions in the k space? (5 pts) (d) Reconcile why it is equivalent to evaluate p ˆ in the x or k space for any arbitrary wavefunction ψ (x,t) , i.e., k x A p A p | ˆ | |...

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