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Fall 2007 2 Mech

Fall 2007 3 EM

Fall 2007 4 EM

Fall 2007 5 QM

Fall 2007 6 QM

Fall 2007 7 SM

Fall 2007 8 SM

Fall 2007 9 Misc

Fall 2007 10 Misc

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Fall 2007 13 EM

Fall 2007 14 EM

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Fall 2007 20 Misc

A system is composed of *N* identical classical oscillators, each of mass *m*, defined on a one-dimensional lattice. The potential for the oscillators has the form

(Thus, the oscillators are harmonic for *n* = 2 and anharmonic otherwise).

Find the average thermal energy at temperature *T*.

Hint: An integral that appears in the course of evaluating the partition function cannot be computed in terms of elementary functions. Fortunately,it amounts only to an unimportant overall coefficient.

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Answer

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Classical partition function for a single oscillator is

(2)By change of variables, we bring this to the form

(3)where

(4)A learned reader would recognize this as the Euler Gamma-function. However, knowing this is not necessary. The product $\Gamma(1/2)\Gamma(1/n)$ is just a numerical coefficient, which will disappear from the final result.

The average energy is given by

(5)This result resembles the equipartition theorem in the sense that material constants do not enter.

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