A thermodynamic analysis of the harmonic oscillator is presented. The motivation is provided by the blackbody radiation spectrum, because radiation modes take the harmonic-oscillator form. We use the behavior of a thermal harmonic oscillator system under a quasistatic change of oscillator frequency ω to show that the thermodynamic functions can all be derived from a single function of analogous to Wien’s displacement theorem. The high- and low-frequency limits yield asymptotic forms involving the temperature alone or frequency ω alone, corresponding to energy equipartition and zero-point energy. We suggest a natural interpolation between the limiting forms. The Planck spectrum with zero-point energy corresponds to the function satisfying the Wien displacement result which provides the smoothest possible interpolation between energy equipartition at low frequency and zero-point energy at high frequency.
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I do not know of any text where the harmonic oscillator is treated as a thermodynamics system apart from statistical mechanics.
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I am not aware of any thermodynamic treatment of the quasistatic change of the harmonic oscillator frequency. The familiar derivations of the Wien law take the Doppler-shift form given in Ref. 3.
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13.
I am not aware of any textbook treatment of blackbody radiation that mentions zero-point energy as part of the spectrum involved.
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