High Order Perturbation Theory for a Generalized Central Field Perturbation of the Three-Dimensional Harmonic Oscillator

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Abstract
A method is presented to solve a general even‐order perturbation of the three‐dimensional harmonic oscillator to any order of perturbation. It is found that using an iterative process it is possible to express the nth order perturbation wavefunction as a sum of 4n terms. These 4n terms depend upon all the lower order perturbation wavefunctions and energy corrections, and may be easily evaluated. It is also shown that accurate results may be achieved for large perturbations if the energy is re‐expressed as a Pade approximant. The nine lowest levels of an oscillator perturbed by a quartic term are calculated to tenth order and the results agree qualitatively with the two dimensional mixed harmonic—quartic oscillator results of Bell, Davidson, and Warsop.