Energies of oscillators with mixed quartic and sextic anharmonicities

Abstract

Energy values for different states of oscillators with quartic, sextic and mixed quartic and sextic anharmonicities have been calculated employing the renormalized hypervirial perturbation series. The K parameter, which yields the best convergent series, has been found to be different from the one derived from the analytic approach put forward by Chaudhuri and Mondal. The accuracy of the values obtained and the dependence of K on the perturbation parameters and the quantum numbers of the state have been discussed for all the cases.

Determination of free energies of an oscillator with mixed quartic and sextic anharmonicities

“…This is the same equation as obtained from the approach suggested by Chaudhuri and Mondal [24] to get the renormalization parameter for the determination of energy eigenvalues of the oscillator with mixed quartic and sextic anharmonicities [22] using Killingbeck's technique [25]. So we can expect the results of the BF method to be as good as the energies calculated by employing the above referred to parameter.…”

“…So we can expect the results of the BF method to be as good as the energies calculated by employing the above referred to parameter. It has been found [22] that this approach fails for large anharmonicities and the renormalization parameter has to be treated as a variational quantity. Besides, for finite P, contributions to F from the higher-n energy states become significant and their importance increases with a rise in temperature.…”

“…Recently, the scope of the formulation given in Ref. [22] has been enlarged to get an n-dependent renorrnalization parameter by using second-order perturbation theory, and it has been concluded that this does improve upon the results for energy eigenvalues, but still the accuracy is very poor [26]. On this basis, it is felt that looking for an improvement through such an approach will be futile.…”

“…In this work we compare the techniques of [14] and [20] for the oscillators with quartic (g=0), sextic (a=0), and mixed quartic-sextic (a, pl&0) anharmonicity for wide range of a, q, and P values and bring out the dependence of F on these. On testing the results against the exact values found by summation of the Boltzmann factors for the energy eigenvalues of these oscillators [21,22], it is concluded that for them the approximate procedure given by Feynman and Kleinert [14] is more accurate than that of Biittner and Flytzanis [20].…”

“…In the framework of the Buttner-Flytzanis approach [20], we compare the actual AHO with a reference HO of frequency co' with its center displaced by b and select the parameters co' and b in such a manner that the expression for the free energy, including terms up to first order in perturbation, given by [21,22] or obtained by employing the technique described in Ref. [22].…”