Physics 424: Quantum Mechanics, Fall 2015Reading/Homework Assignment 10Due in Class Friday, 12/4/151Reading•Townsend Chapter 9 - Translational and Rotational Invariance. We willskip the following sections and you do not need them for the homeworkor exams:–Sec. 9.2 – Translational Invariance.–Sec. 9.3 – COM Coordinates.–Sec. 9.7 – Diatomic Molecules.You should read all of Ch. 9 apart from the three sections listed above.•Start reading Townsend Chapter 10 – Bound State of Central Poten-tials. You don’t need Ch. 10 for this homework set, but we will startdiscussing it in class.Problems to work out and turn in (60 pts total)1. (10 pts) Using the position-space representation of the position andmomentum operators in 3Dˆx|ψi ↔xψ,ˆy|ψi ↔yψ . . .ˆpx|ψi ↔ -i¯h∂ψ∂x,ˆpy|ψi ↔ -i¯h∂ψ∂y. . .(1)prove the following canonical commutation relations[ˆxj,ˆpk] =i¯hδjk,ˆI[ˆxj,ˆxk] =ˆ0,[ˆpj,ˆpk] =ˆ0.(2)Do this by applying the operators of Eq. 1 to an arbitary wave functionψ(x, y, z). Here we are using the same notation as Townsend Sec. 9.1,so for example ˆx2= ˆy. Sincejandkcan each be 1, 2, or 3 there are inprinciple 27 commutation relations to show in Eq. 2. However, it willsuffice for you to prove a much smaller number of examples and thenargue that the remaining relations are proved in the same way.
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