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PHZ 3113 Fall 2011 – Homework 10 Due at the start of class on Friday, November 18. Half credit will be available for homework submitted after the deadline but turned no later than the start of class on Monday, November 21. Answer all questions. Please write neatly and include your name on the front page of your answers. You must also clearly identify all your collaborators on this assignment. To gain maximum credit you should explain your reasoning and show all working. 1. Find the eigenvalues and corresponding orthonormal eigenvectors of the following symmetric or Hermitian matrices. (Do this by hand; don’t just plug the matrices into a calculator or computer program.) √ 1 8 √0 0 1 1 √ 3 2 2 1−i (a) (b) (c) 8 √1 8 (d) 1 0 1 2 0 1+i 3 1 1 0 0 8 1 2. Find the eigenvalues e1 , e2 and corresponding normalized eigenvectors v1 , v2 of the non-symmetric matrix −1 −2 . 4 3 Check whether the eigenvectors are orthogonal in the sense that v1† · v2 = 0. 3. (A generalization of Example 3 on page 165 of Boas.) Consider two identical pointlike objects, each of mass m, resting on a horizontal, frictionless surface between two rigid walls. The masses are connected to each other by a spring of spring constant k. Then each mass is connected to a different wall by a spring of spring constant k 0 in such a way that all three springs are stretched and motion takes place along a single straight line. (a) Write down the equation of motion for the displacement of each mass from its equilibrium position. (b) Write down and solve the characteristic equation for the eigenfrequencies of this system. (c) Find the eigenvectors and hence the normal modes of the system.