3063_Exam3_sp06 - PHY3063 Spring 2006 R. D. Field PHY 3063...

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PHY3063 Spring 2006 R. D. Field Final Exam Page 1 of 7 April 25, 2006 PHY 3063 Final Exam Name_________________ Problem 1 (35 points): Consider an particle with mass m confined within an infinite square well defined by V(x) = 0 for 0 < x < L , V(x) = + otherwise. (a) (5 points): Using Schrödinger’s equation calculate the allowed stationary state eigenfunctions ψ n (x) , where the complete wavefunctions are given by h / ) ( ) , ( t iE n n n e x t x = Ψ ψ . Normalize the eigenfunctions so that the probability of finding the particle somewhere in the box is one. (b) (5 points): Show that the allowed energy levels of the system are, E n = E 0 n 2 , where ) 2 /( 2 2 2 0 mL E π h = is the ground state energy and n = 1, 2, 3, …. Why is n = 0 excluded as a possible energy level? (c) (10 points): Consider the operator, O = (x) op (p x ) op ( i.e. the product of the position operator times the momentum operator). Is the operator O hermitian? Calculate the expectation value of the operator O for the n th stationary state ( i.e. calculate < ψ n |O| ψ n >). (d) (15 points): Suppose the particle in this infinite square well has an initial wave function at t = 0 given by ) / ( sin ) 0 , ( 2 L x A x = Ψ . What is the normalization A ? If you measure the energy of this particle, what is the probability that you will measure the ground state energy E 0 ? V = +infinity V = +infinity Infinite Square Well 0 L x
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PHY3063 Spring 2006 R. D. Field Final Exam Page 2 of 7 April 25, 2006 PHY 3063 Final Exam Name_________________ Problem 1 Scratch Paper
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PHY3063 Spring 2006 R. D. Field Final Exam Page 3 of 7 April 25, 2006 Problem 2 (35 points): Consider the case of two non-interacting particles both with mass m
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This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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3063_Exam3_sp06 - PHY3063 Spring 2006 R. D. Field PHY 3063...

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