对微扰论波函数的非正交修正
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摘要
不含时微扰论对非简并能级的修正是相当精确的,然而对波函数的修正精度却不能令人满意.经过审视微扰论的推导过程,可以发现,造成这一精度差异的原因或许就是"正交性假设"."正交性假设",即零阶以上的任意阶修正波函数与零阶波函数都正交,是建立微扰论的过程中习惯上使用的一个附加条件.详细探讨了"正交性假设",并利用波函数的归一化属性得到了关于高阶修正波函数的一个约束条件,而这个条件暗示了在二阶及以上精度不适合继续使用"正交性假设".可以证明,在不引入"正交性假设"的情况下,能级修正的结果和正交情况是完全一致的,但是修正波函数的结果与正交情况却出现了不容忽视的差异.这个现象可以合理解释之前的精度问题.作为一个具体示例,计算了匀强电场中一维带电谐振子系统的前三阶非正交修正波函数.对比此系统的解析解,可以发现波函数的非正交修正比正交修正确实具有更高的精度.简单探讨了推广到简并微扰论的情况,结合近期Stark问题的进展,给出了检验非正交微扰修正的思路.
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.
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.
引文
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[2]KATO T. Perturbation Theory for Linear Operators[M]. Berlin:Springer Science&Business Media, 2013.
[3] LIONS J L. Exact Controllability, stabilization and perturbations for distributed systems[J]. SIAM Review,1988, 30(1):1-68.
[4]曾谨言.量子力学教程[M].北京:科学出版社,2014.
[5]曾谨言.量子力学(卷I)[M].北京:科学出版社,2013.
[6]钱伯初.量子力学[M].北京:高等教育出版社,2006.
[7] SAKURAI J J, COMMINS E D. Modern Quantum Mechanics[M]. New York:Addison-Wesley Publishing Company, Inc, 1985.
[8]COHEN-TANNOUDJI C, DIU B, LALOE F. Quantum Mechanics(vol.2)[M]. Hoboken:John Wiley&Sons Inc,1991.
[9]梁灿彬,周彬.微分几何入门与广义相对论(下册)[M].北京:科学出版社,2006.
[10] GRIFFITHS D J. Introduction to Quantum Mechanics[M]. Cambridge:Cambridge University Press, 2016.224-225.
[11]汤川秀树.量子力学I[M].北京:科学出版社,1991.
[12] OSHEROV V I, USHAKOV V G. Stark problem in terms of the Stokes multipliers for the triconfluent Heun equation[J]. Physical Review A, 2013, 88(5):053414.
[13] OSHEROV V I, USHAKOV V G. Analytical solutions of the Schr(o|¨)dinger equation for a hydrogen atom in a uniform electric field[J]. Physical Review A, 2017, 95(2):023419.
[14] WEINBERG S. The Quantum Theory of Fields(vol.1)[M]. Cambridge:Cambridge University Press, 1995.