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HW 5 Ch03.5 and 4

# HW 5 Ch03.5 and 4 - Phys 404 Spring 2010 Homework 5 CHAPTER...

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Phys 404 Spring 2010 Homework 5, CHAPTER 5 Due Thursday, March 25, 2010 @ 12:30 PM General hint : Problems 1 – 3 involve classical statistical mechanics. In this case, the sums over quantum states are replaced by integrals over all position and momentum components, so that, for example, ( ) ( ) -H(p,q) NN Nd 1 ... X(p,q)e dp dq h X= Z   τ  ∫∫  ( ) ( ) -H(p,q) Nd 1 Z = ... e dp dq h τ Here N is the number of particles, d is the spatial dimensionality of the problem, H( p , q ) is the Hamiltonian (total energy) in terms of the vector coordinates ( q ) and momenta ( p ) of all the particles, and h is Planck’s constant. Note that there is one integral for each component of momentum and position for each particle, so that there are 2Nd integral signs indicated by the ... in the formulas. See the Lecture 13 summary for a review of classical statistical mechanics. 1 . A pendulum hangs under gravity, has length L and mass m , and makes an angle θ with the vertical direction. Assuming that the amplitude of oscillation is small, find < >, < 2
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