Journal of East China Normal University(Natural Sc ›› 2019, Vol. 2019 ›› Issue (2): 97-105,115.doi: 10.3969/j.issn.1000-5641.2019.02.010

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Non-orthogonal corrections to wave functions in perturbation theory

SHEN Jia-yin, XUE Xun   

  1. School of Physics and Materials Science, East China Normal University, Shanghai 200241, China
  • Received:2018-03-06 Online:2019-03-25 Published:2019-03-27

Abstract: Time-independent perturbation theory is fairly accurate for the correction of non-degenerate energy levels, but its accuracy is not satisfactory for the correction of wave functions. After examining the derivation process of perturbation theory, it was found that the reason for the difference in precision may be related to the Orthogonality Assumption. The Orthogonality Assumption-an arbitrary-order modified wave function above zero order is orthogonal to the zero-order wave function-is a condition used in establishing perturbation theory. This paper explored the Orthogonality Assumption in detail and obtained a constraint condition on higher-order modified wave functions by using the normalized properties of the wave function; this condition implies that the accuracy at second-order and above is not suitable for use with the Orthogonality Assumption. It can be shown that without introducing the Orthogonality Assumption, the result of the energy level correction is exactly the same as that of the orthogonal situation, but the result of the modified wave function has a difference that cannot be ignored. This phenomenon can reasonably explain the previous accuracy problem. In this paper, the first three-order non-orthogonal corrective wave function of the one-dimensional charged harmonic oscillator system in the homogeneous electric field is taken as a specific example. By comparing the analytical solution of this system, it can be demonstarted that the non-orthogonal correction of the wave function has higher accuracy than the orthogonal correction. The paper briefly discusses generalization to the degenerate perturbation theory. Combined with recent progress on the Stark problem, it offers a possible method to check for correction of non-orthogonal perturbation.

Key words: time-independent perturbation theory, non-degenerate correct energy level, correct wave function, orthogonality assumption

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