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Unformatted text preview: ECE 3060 Fall 2008 Prelim Exam 1 Solution Rules of the Exam (Please read carefully before start) 1. This is an open-book, open-note exam. You are allowed to use your computer as a browser for downloaded course files, but you are NOT allowed to connect to Internet in any form. Connection to Internet during the exam period will be considered as a violation of academic integrity. 2. Grading will ONLY consider what you legibly put down on the exam paper. References to textbook or class notes will NOT count for credit. Irrelevant answers, even though the content is correct, will NOT receive any partial credit. Wrong information will always cause a deduction in total credit. 3. The time for the exam will be exactly 90 minutes. Do not be trapped in a question you cannot answer, and use your time wisely for distributing your efforts in different problems. Do not diverge into irrelevant answers, since this will negatively impact your performance. 1. Determine if the following operators are Hermitian. You need to clearly state your reasons. (a) x i p - = (3 pts) Hermitian, it is observable. You can use definition as well. (b) x i p k - = = (3 pts) A real constant times a Hermitian operator is always Hermitian. (c) k (4 pts) Since we know k i x = and is Hermitian, hence k cannot be Hermitian. (d) log (x/x ) (4 pts) Any real functions of x will be Hermitian. However, there is a singularity at 0 for this function. 2. What are the eigenfunctions and the corresponding eigenvalues for the following operators? (a) The momentum operator x i p - = in x space. (4 pts) The eigenfunction is e ikx , and the corresponding eigenvalue is k i . This can be directly solved from the eigenvalue equation....
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This note was uploaded on 11/07/2009 for the course ECE 4060 at Cornell University (Engineering School).