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Unformatted text preview: PHYS 333 Winter 2009 Assignment 9 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. The approach is to pretend that the particle’s position takes on discrete values, so that its potential energy also has discrete values. Then the partition function can be written as the usual sum over states. Z 1 = X i e- u ( x i ) /k B T and the average value of the position becomes ¯ x = 1 Z 1 X i x i e- u ( x i ) /k B T In the limit where the two sums becomes integrals: ¯ x = R ∞-∞ xe- u ( x ) /k B T dx R ∞-∞ e- u ( x ) /k B T dx as required. 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 : u ( x ) = u ( x ) + 1 2 ( x- x ) 2 d 2 u dx 2 x + 1 3! ( x- x ) 3 d 3 u dx 3 x + ··· 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 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 3 u dx 3 . You will need to use the integrals Z ∞-∞ e- βay 2 dy = r π βa 1 and Z ∞-∞ y 4 e- βay 2 dy = 3 4 β 2 a 2 r π βa Write the Boltzmann factor as e- βu ( x ) = e- β [ u ( x )+ ay 2 + by 3 ] where y = x- x . The expansion has no linear term because u ( x ) is a minimum. Next, write e- βby 3 as e- βby 3 ' 1- βby 3 so that ¯ x becomes ¯ x ' e- u ( x ) /k B T R ∞-∞ [ x- βby 4 ] e- βay 2 /k B T dy e- u ( x ) /k B T R ∞-∞ e- βay 2 /k B T dy The other terms in the two integrals, those that depend on odd powers of y , have been omitted: they are necessarily zero. The e- u ( x ) /k B T factors cancel, and we arrive at ¯ x ' x- 3 4 b βa 2 = x- 3 b 4 a 2 k B T Because of the y 3 term, the average position changes with temperature....
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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.
- Winter '09