# hw6 - PHY 4604 Fall 2010 Homework 6 Due at 5:00 p.m on...

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PHY 4604 Fall 2010 – Homework 6 Due at 5:00 p.m. on Friday, November 12. Please turn in your homework in class before the deadline, or else bring it to NPB 2162. (You may push it under the door if NPB 2162 is unoccupied.) No credit will be available for homework submitted after the start of class on Monday, November 15. Answer all four questions. Please write neatly and include your name on the front page of your answers. You must also clearly identify all your collaborators on this assignment. To gain maximum credit you should explain your reasoning and show all working. 1. Compatible observables. Two observables Ω and Λ are said to be “compatible” if the corresponding operators commute, i.e., [ ˆ Ω , ˆ Λ] = 0. Compatible observables can be chosen to have a complete set of simultaneous eigenstates {| ω, λ i} such that ˆ Ω | i = ω | i and ˆ Λ | i = λ | i . Example (based on Problem 1-23 in Modern Quantum Mechanics by J. J. Sakurai): Consider a three-dimensional Hilbert space. In a certain basis, operators ˆ Ωand ˆ Λare represented by ˆ Ω= a 00 0 - a 0 - a , ˆ Λ= b - ib 0 ib 0 , where a and b are both real. (a) Show that ˆ ˆ Λcommute . (b) Find the spectrum of ˆ Ω and the spectrum of ˆ Λ. (c) Find a complete orthonormal set of simultaneous eigenstates of ˆ ˆ Λ. Label each eigenstate according to its eigenvalues of ˆ ˆ Λ.

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hw6 - PHY 4604 Fall 2010 Homework 6 Due at 5:00 p.m on...

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