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Internal Energy of N Anharmonic Oscillators

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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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