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CONVERGENT PERTURBATION EXPANSION FOR THE ANHARMONIC OSCILLATOR (1982)
BibTeX
@MISC{Au82convergentperturbation,
author = {C. K. Au and G. W. Rogers and Y. Aharonov},
title = {CONVERGENT PERTURBATION EXPANSION FOR THE ANHARMONIC OSCILLATOR},
year = {1982}
}
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Abstract
We study the ground state as well as the first three excited states of the anharmonic oscillator with anharmonicity hx 4 for a range of h = (0, 10) with the first-order logarithmic perturbation iteration method (FOLPIM). This leads to conver-gent results. The initial choice of the wave function seems only to affect the rate of convergence in the case of the ground state but may critically affect the convergence for the excited states. For large values of h, convergence is best obtained by choosing the asymptotic solution as the initial "unperturbed " wave function. The perturbative solution of the ground state of the anharmonic oscillator has been extensively studied [1-23]. In the typical problem where the anharmoni-ticity is Xx 4, the energy corrections have been calcu-lated by Bender and Wu to 150 orders [4]. The asymp-totic form for the energy correction as a function of the order of the perturbation parameter X has also been given [4]. The resultant series is well known to be divergent. Both the Pad6 approximation and the Borel summation method have been used to recover finite results for the energy correction [13,14,18]. In the past few years, there has been a lot of renewed in-terest in the logarithmic perturbation theory which es-sentiaUy consists of transforming the Schr6dinger equation to Ricatti form by taking the logarithm of the wave function [8-11,24-28]. For the case of the ground state where the wave function does not con-tain any zero, the logarithm is regular. For excited states where the wave functions have nodes, the zeros must be factored out before the logartihm is taken of the envelope of the wave function [24]. The logarithmic perturbation expansion has been used to restudy the?,x 4 anharmonic oscillator to 72nd
Keyphrases
anharmonic oscillator wave function ground state excited state convergent perturbation expansion energy correction finite result conver-gent result resultant series perturbative solution asymp-totic form borel summation method pad6 approximation asymptotic solution unperturbed quot past year perturbation parameter large value initial choice logarithmic perturbation theory initial quot ricatti form logarithmic perturbation expansion typical problem studied 1-23 first-order logarithmic perturbation iteration method es-sentiauy consists schr6dinger equation
