量子力学中的微扰论

Perturbation methods in quantum mechanics

简要介绍了量子力学中三种定态微扰方法 :一是通常的Rayleigh -Schr¨odinger (R -S)微扰 ,但其缺点是高阶微扰项的表达式很繁 ,且无一般的通项公式 .这里介绍了一个能量和波函数的递推公式 ,便于计算机编程计算 .二是由Brillouin -Wigner (B -W)微扰论演变出的迭代微扰法 ,对于有限维不变子空间或在有限维空间截断近似下计算 ,收敛是很快的 ,收敛范围大于R -S法 ,最便于上机计算 .三是近年来提出的超收敛微扰法 ,它类似于经典力学中的超收敛的Kolmogorov微扰方法 ,有很大的理论意义 .本文给出一种更简单明了的推导 .

Three kinds of perturbation methods for nondegenerate quantum system are briefly introduced and compared applying to a simple model.The first one is Rayleigh-Schrdinger (R-S) method which is widely discussed in text book,a very useful recurrence relation is also given.The second one is iterative perturbation method which is formulated based on Brillouin-Wigner (B-W) perturbation,it can be used to make a program to evaluate the eigensolutions by computer. The third one is superconvergent method proposed by Scherer,which is important to determine the existence of quantum Lie transformation.