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hw2246sol - Physics 246 Spring 2007 Homework#2 Due in class...

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Physics 246, Spring 2007 Homework #2 Due in class, Wednesday, February 7, 2007 Feel free to discuss the problems with me and/or each other. Each student must write up his/her own solutions separately. Each problem is worth 10 points unless otherwise indicated. 1. Liboff, problem 3.17, p. 87. 2. a) Liboff, problem 3.20, p. 87, part a. b) Now take x 1 = x and x 2 = y , with ˆ H = 1 2 m p 2 x + 1 2 m p 2 y . What is ψ ( x, y ) and what are ψ 1 ( x ) and ψ 2 ( y ) in this case? 3. (15 points) A system at t = 0 is in the state ψ ( x, 0) = 1 ( x ) + 2 ( x ) where φ 1 ( x ) and φ 2 ( x ) are real eigenfunctions of the Hamiltonian with eigenvalues E 1 and E 2 and A, B are real. a) Assuming that φ 1 ( x ) and φ 2 ( x ) are properly normalized, what values can A and B take on? (Hint: R φ * 1 φ 2 dx = 0.) b) Show that the probability distribution is of the form P ( x, t ) = f ( x ) + g ( x ) cos ωt . c) What are f ( x ), g ( x ), and ω ? d) For what allowed values of A, B is g ( x ) maximized? 4. (15 points) A bead of mass m is constrained to move on a frictionless wire bent into a circle of radius a .
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