Phys 404 Spring 2010 Homework 5,CHAPTER 5Due 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)NNNd1... X(p,q)edpdqhX=Zτ∫∫( ) ( )-H(p,q)Nd1Z =... edpdqhτHere 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 summaryfor a review of classical statistical mechanics. 1. A pendulum hangs under gravity, has length Land mass m, and makes an angle θwith the vertical direction. Assuming that the amplitude of oscillation is small, find <>, <2
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