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Unformatted text preview: 1 ECE 4060 Fall 2009 Prelim Exam 2 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 most often cause a deduction in total credit. 3. The time for the exam will be limited. 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. An atom at a mass of 10-25 kg is inside a simple harmonic oscillator described by the Hamiltonian of 2 2 2 2 2 2 1 2 ˆ x m dx d m H ω + − = h where ω is 10 14 Hz. (a) How many eigenstates have energy below 1eV? (3 pts) Using eV is the easiest here: eV n 1 2 1 < ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ω h ; n ≤ 14. There are 15 eigenstates. (b) If the atom is at the ground state, what is the uncertainty in the position (in cm)? What is the classical turning point (in cm)? (4 pts) Due to the use of mass in kg, everything has to be in MKS. m m x m m x turn 12 12 10 2 . 3 10 3 . 2 2 − − × = = × = = Δ ω ω h h The sub-atomic precision in this SHO is expected since the atom has significant mass. (c) If the atom at t = 0 is at the superposition state of 1 5 3 5 4 φ φ + , find E and Δ x . (8 pts) eV J E E E 056 . 10 . 9 50 43 5 3 5 4 21 1 2 2 = × = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − ω h . You can use two ways to estimate Δ x : ( ) 2 2 x m E Δ ≅ ω or you will need to go through the promoter and demoter. You will obtain Δ x ∼ 3.0×10-12 m. 2 (d) Following (c), find the expression for ( ) t x (4 pts) ( ) ( ) t m a a m t x cos 2 25 24 | | 2 ω ω ψ ψ ω h h ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + = ⊥ (e) If instead the atom at t = 0 is at the superposition state of 2 5 3 5 4 φ φ + , find the expression for ( ) t x (3 pts) The index is off by 2, and both are even functions, so we will have ( ) = t x (f) If the atom is originally ( t < 0 ) at the ground state when ω is 10 14 Hz, and at t = 0 the frequency is changed to ω 1 = 2×10 14 Hz. If we make a measurement of total energy at t = 0 , what are the probabilities that the return values are 2 ω h , ω h and 2 3 ω h ? (8 pts) The new states have eigenenergies of ( )...
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell.