Internal Energy of N Anharmonic Oscillators
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A system is composed of N one dimensional classical oscillators. Assume that the potential for the oscillators contains a small quartic "anharmonic" term
V(x) = (m*Ω^2)/2 + a*x^4
Where a*(x^4) <<< KB T and (x^4) = average value
Calculate the average energy per oscillator to the first order in a
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Solution Summary
Internal energy of N Anharmonic Oscillators are examined. The average energy per oscillator to the first order is determined.
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The partition function for a single particle is given by:
Z1 = Z_{trans} Z_{pot}
Where:
Z_{trans} = 1/h Integral from minus infinity to plus infinity of exp[-beta p^2/(2m)] dp (1)
and
Z_{pot} = Integral from minus infinity to plus infinity of exp[-beta (1/2 m omega^2 x^2 + a x^4)] dx (2)
To evaluate these integrals we can use that:
Integral from minus to plus infinity of exp(-c x^2) dx = sqrt(pi/c) (3)
To evaluate Z_{trans} we only need to insert c = beta/(2m) in (3):
Z_{trans} = 1/h sqrt(2 m pi/beta) = sqrt(2 pi m k T/h^2) ...
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