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homework1 - tions 3 Consider the equation of a damped...

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Homework 1 PHZ 5156 Due Thursday, August 31 1. Consider a particle of mass m trapped in a two-dimensional box with infinitely high walls at x = 0, x = a , y = 0, and y = b . Inside the box, the potential is zero and the Hamiltonian is given by ˆ H = - 2 2 2 m = - 2 2 m 2 ∂x 2 + 2 ∂y 2 (1) The infinitely high potential walls that trap the particle result in the boundary conditions for the wave function ψ ( x = 0 , y ) = 0, ψ ( x = a, y ) = 0, ψ ( x, y = 0) = 0, and ψ ( x, y = b ) = 0. a) Use the method of separation of variables to write the time-independent Schrodinger equation ˆ ( x, y ) = ( x, y ) (2) as two ordinary differential equations. b) Determine the solutions to the two equations obtained in part a). c) What are the allowed energies E ? 2. The torsion of a bar is described by the fourth-order equation, d 4 θ dx 4 + τθ = 0 (3) Show how this can be expressed as a system of coupled first-order differential equa-
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Unformatted text preview: tions. 3. Consider the equation of a damped, driven simple harmonic oscillator, m d 2 y dt 2 + γ dy dt + ky = Fcos ( ωt ) (4) a) Find the particular solution y ( t ) to this equation. Write your answer in the form y ( t ) = | A | cos ( ωt + δ ), and determine expressions for | A | and the phase angle δ . To keep your work simple, use the definition for the natural frequency of the oscillator ω = q k m . 1 b) Determine the condition for resonance. In other words, find an expression for the driving frequency ω which results in a maximum amplitude | A | for the response. 4. Write down first-order finite difference approximations for each of the follow-ing: a) df dx b) d 2 f dx 2 c) d 3 f dx 3 d) ∂ ∂t ∂f ∂x 2...
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