hw01 - Introduction to Quantum and Statistical Mechanics...

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Introduction to Quantum and Statistical Mechanics Homework 1 Prob. 1.1. To establish the intuitive link between the Hamiltonian mechanics and the wave equation, we will use the example of the longitudinal vibrations in classical mechanics written as the wave equation. Consider n equal particles on a line from 0 to L , coupled by identical strings with linear spring constant of k . At equilibrium, all particles are equally spaced at a distance x = L/(n+1) . Let z j be the displacement of the j th particle from equilibrium along the x axis (measured positive to the right). (a) Show that the motions are governed by the differential equations: (10 pts) n j z z z k dt z d m j j j j ,..., 1 ), 2 ( 1 1 2 2 = + - = - + (b) Let n →∞ , m 0, x 0 ; obtain the wave equation as the limit. What is the group velocity now? What is the phase velocity? (10 pts) . (c) What is the role of L in the derivation, assuming fixed boundaries at 0 and L ? Qualitatively describe the vibration behavior when
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hw01 - Introduction to Quantum and Statistical Mechanics...

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