Homework Problems in Quantum Mechanics and Vibrations, Assignments of Physics

Download Homework Problems in Quantum Mechanics and Vibrations and more Assignments Physics in PDF only on Docsity! 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 Ĥ = −~ 2∇2 2m = − ~ 2 2m ( ∂2 ∂x2 + ∂2 ∂y2 ) (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 Ĥψ(x, y) = Eψ(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, d4θ dx4 + τθ = 0 (3) Show how this can be expressed as a system of coupled first-order differential equa- tions. 3. Consider the equation of a damped, driven simple harmonic oscillator, m d2y dt2 + 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 ω0 = √ k m . 1