finalSol08 - ECE 3060 Fall 2008 Final Exam Solution Rules...

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1 ECE 3060 Fall 2008 Final Exam 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 will be considered as 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 2.5 hours. 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. Which of the following is not a direct contribution from Paul A. M. Dirac? (3 pts) (a) The relativistic Schrödinger equation (b) The bracket notation in finite-base approximation (c) The statistics of Fermions (d) Fermi’s golden rule (e) Canonical conjugates in uncertainty principle (such as ( x,p ), ( E,t ), etc.) (f) Quantization of angular momentum in hydrogen atom Ans: (f) , which is by Bohr. 2. What are the Fourier transforms of the following functions in the reciprocal k space? No derivation or normalization is necessary. (a) The plane wave function e ikx (3 pts) The delta function: δ (k) (b) The Gaussian function exp( - α x 2 ) (3 pts) The Gaussian function exp( -k 2 /4 ) (c) The wave packet f(x) exp( ik 0 x ) given that the Fourier transform of f(x) is F(k) (3 pts) A shift in k space, thus: F(k – k 0 ) . 3. If the Hamiltonian in a given system is known, how will you check if a physical observable conserves in time for its expectation value? (4 pts) From the Erhenfest’s theorem: > < + > < >= < t Q H Q i Q dt d ˆ ] ˆ , ˆ [ 1 ˆ h , if the Hermitian operator corresponding to the physical observable has no explicit time dependence, then the physical quantity conserves in time if it commutes with the Hamiltonian.
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2 4. The eigenstates of the H-atom Hamiltonian are given by, ( )( ) ( ) φ θ ψ , , , , , , m l l n m l n Y r R r = .
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finalSol08 - ECE 3060 Fall 2008 Final Exam Solution Rules...

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