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finalSol09 - ECE 4060 Fall 2009 Final Exam Solution Rules...

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1 ECE 4060 Fall 2009 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 does not directly relate to the Plank constant h ? There can be no, one or multiple answers. (3 pts) (a) The spectrum of black-body radiation. (b) The quantum noise in the simple LC circuit. (c) The Uncertainty Principle governing Δ E and Δ t . (d) Plank’s assumption on the entropy of a quantum system. (e) The de Broglie wave-particle duality. (f) The zero-point energy in a simple harmonic oscillator. Answer: (d), which applies to the classical mechanics as well. 2. Two wavefunctions describe the electron in a hydrogen atom: 321 1 31 200 211 201 2 1 2 1 2 1 ; 2 1 2 1 ψ ψ ψ ψ ψ ψ ψ + + = + = B A . The quantum numbers represent ( ) ( ) ( ) φ θ φ θ ψ , , , , lm l n nlm Y r R r = . For the following operators, describe whether these TWO functions are eigenfunctions. If yes, give the corresponding eigenvalues; if no, give the expectation values. (a) The Hamiltonian H (6 pts) ψ A ψ B Eigenfunction of H? Yes No Eigenvalue or H -E 0 /4 = -3.4eV eV E E E 98 . 1 48 7 9 2 1 4 1 4 4 1 0 0 0 = = + +
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2 (b) The angular momentum L z (6 pts) ψ A ψ B Eigenfunction of L z ? Yes No Eigenvalue or Z L h ( ) 4 2 1 1 4 1 h h h = + (c) The amplitude of the total angular momentum L 2 (6 pts) ψ A ψ B Eigenfunction of L 2 ? No No Eigenvalue or 2 L 2 2 2 1 2 1 0 2 1 h h = + 2 2 2 7 3 2 2 1 2 1 4 1 0 4 1 h h = + + 3. Which of the following system has the density of states nearly independent of E ? There can be no, one or multiple answers. (4 pts) (a) The 1D harmonic oscillator (b) The 2D particle-in-a-box (c) The 3D particle-in-a-box (d) The potential in a 3D system that has potential in the x direction as quadratic well and in the y and z directions as particle-in-a-box (e) The potential in a 3D system that has potential in the x and y direction as quadratic wells and in the z direction as particle-in-a-box (f) The hydrogen atom (g) The phonons in a 3D lattice Ans: (a) and (b). You have done most of these DOS in the examples and homework.
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