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newassign5 - Problem Set 05 Note: The following problem set...

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Problem Set 05 Note: The following problem set is due October 10 by midnight. Please return directly to me in my oFce Rutherford 321. If I’m not there slip your assignment below my door. 1. Give short answers to the following questions: (a) Using matrices show that | ψ ih ψ | is an operator whereas h ψ | ψ i is a number. (b) Consider two eigenvectors | ψ i and | ϕ i of an operator A . Show that these eigenvectors are orthogonal, i.e h ψ | ϕ i = 0. (c) If two operators A and B commute, and if | ψ i and | ϕ i are two eigenvectors of A with diFerent eigenvalues, then show that h ψ | B | ϕ i = 0. (d) If | φ i and | ψ i are any two vectors in a Hilbert space H , show that: |h φ | ψ i| 2 ≤ h φ | φ ih ψ | ψ i where the equality is realised only if the two vectors are proportional to each other. (e) Two operators A and B do not commute and their commutator is given by a c -number (i.e a constant): [ A , B ] = b where b is a constant. Show that: [ A , F ( B )] = bF 0 ( B
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This note was uploaded on 12/15/2010 for the course PHYS 357 taught by Professor Keshavdasgupta during the Fall '05 term at McGill.

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newassign5 - Problem Set 05 Note: The following problem set...

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