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    Heat capacity of a system of harmonic oscillators

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    The Hamiltonian of the one dimensional harmonic oscillator is:

    H = p^2/(2m) + 1/2 m omega^2 x^2

    The partition function in the classical regime can be computed as follows. We use that the number of quantum states in a range dp of momentum space and dx in configuration space is dpdx/h. The partition function for a single oscillator can then be written as:

    Z1 = Integral over space and momentum of dpdx/h exp[-beta(p^2/(2m) + 1/2 m omega^2 x^2)] =

    Integral over p from minus infinity to plus infinity of dp/h exp[-beta p^2/(2m)] times

    Integral over x from minus to plus infinity of dx exp(-beta 1/2 m omega^2 x^2)

    We can evaluate these integrals by using:

    Integral from minus to plus infinity of exp(-s t^2)dt = sqrt(pi/s)

    We then get:

    Integral over p from minus infinity to plus infinity of dp/h exp[-beta p^2/(2m)] =

    1/h sqrt[2 m pi/beta]

    And

    Integral over x from minus to plus infinity of dx exp(-beta 1/2 m omega^2 x^2) =

    1/omega sqrt[2 pi/(m beta)]

    The partition function is thus given by:

    Z1 = 1/(h omega) sqrt[2 m pi/beta] sqrt[2 pi/(m beta)] = 2 pi/(h omega beta) = 1/(beta hbar omega) =

    k T/(hbar omega)

    The partition function for N distinguishable oscillators is:

    Z = Z1^N

    The free energy is:

    F = - k T ...

    Solution Summary

    The problem is worked out in detail from first principles.

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