hw9 - PHYS 333 Winter 2009 Assignment 9 Due 5:00 pm Friday...

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PHYS 333 Winter 2009 Assignment 9 Due: 5:00 pm Friday April 3 1. Schroeder, 6.32, modified. Consider a classical particle moving in a one-dimensional potential well u ( x ), as shown in the figure. The particle is in thermal equilibrium with a reservoir at temperature T , so the probabilities of its various states are determined by Boltzmann statistics. (a) Show that the average position of the particle is given by ¯ x = R xe - βu ( x ) dx R e - βu ( x ) dx where each integral is over the entire x -axis. If the temperature is reasonably low, but still high enough for classical mechanics to apply, the particle will spend most of its time near the bottom of the well. In that case we can expand u ( x ) in a Taylor series about the equilibrium point x 0 : u ( x ) = u ( x 0 ) + 1 2 ( x - x 0 ) 2 d 2 u dx 2 ± ± ± ± ± x 0 + 1 3! ( x - x 0 ) 3 d 3 u dx 3 ± ± ± ± ± x 0 + · · · because the linear term must be zero. (b) When the cubic term is a small correction, expand its exponential in (another) Taylor series, keep only the smallest temperature-dependent term, and show that in this limit ¯ x differs from x 0 by a term proportional to k B T . Show that the coefficient of this term is 3 4 b a 2 , where a = 1 2 d 2 u dx 2 and b = 1 3! d
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This note was uploaded on 04/11/2010 for the course PHYS 333 taught by Professor Harris during the Winter '09 term at McGill.

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hw9 - PHYS 333 Winter 2009 Assignment 9 Due 5:00 pm Friday...

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