ps05 - Homework for Physics 316 Modern Physics...

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Unformatted text preview: Homework for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 03/04/05 - 9:55 in 132 Rockefeller Hall Exercise 1: The oscillation amplitude x of a damped classical harmonic oscillator with the force- Cx- b d dt x and the mass m satisfies the differential equation ( m d 2 dt 2 + b d dt ) x =- Cx. (1) This is an eigenvalue equation for the operator on the left hand side. Find the solution x ( t ) for the starting conditions x (0) = x and d dt x (0) = ˙ x . Why is there a solution for all eigenvalues- C whereas there is only a set of discrete eigenvalues for the Schr¨ odinger equation of the Harmonic oscillator potential? Exercise 2: Consider the two dimensional wave equation for the vibration of a rectangular membrane, ∂ 2 z ∂x 2 + ∂ 2 z ∂y 2 = 1 v 2 ∂ 2 z ∂t 2 . (2) The membrane has a boundary at x = 0, y = 0 and x = L x , y = L y at which it does not oscillate, i.e....
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