On the statistical mechanics of quartic type anharmonic oscillators, application of variational and perturbation methods

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Abstract
The lower and upper bounds including variation of the frequency are calculated for the partition function of oscillators of the quartic type. They are shown to be fair approximations to the partition function calculated from eigenvalues given by Hioe and Montroll (1975). The upper bound presents a very good high-temperature approximation which can be differentiated analytically giving also closed formulae for the thermodynamic functions H, E, S, Cp. Perturbation theory is discussed in terms of operator-free generating functions. As expected, it works well only for intermediate temperatures and for small anharmonicity. Using a technique by Fisher (1965) for convex functions, upper and lower bounds for the entropy are also calculated from the exact upper and lower bounds to the partition function.