一种非线性变换及其在量子系统中的应用  

A Nonlinear Transformation and IT'S Applications in Quantum Systems

作　　者：于熙龄[1] 

机构地区：[1]辽宁大学物理学院

出　　处：《辽宁大学学报（自然科学版）》2009年第4期289-309,共21页Journal of Liaoning University：Natural Sciences Edition

摘　　要：给出了一种利用非线性变换(本征映射)求解本征值问题的方法.作为应用的例子,利用这种方法处理了量子力学中的本征值问题.基本思想是:通过引入可调小参数和建立矩阵元方程,把用矩阵表示的算符对角化.根据所处理的问题是否存在扰动的一级近似,将问题分为两种类型;再结合考虑简并、非简并的不同情况,全面、详尽地给出了求解(或近似求解)本征值和本征矢的方法、步骤和有关公式.列举了几个常见的例子,并且尽可能地将所得到的结果与用其他近似方法(例如微扰论、变分法等)得出的结果进行比较,比较的结果表明这个新的方法具有一定的优越性.In this paper we propose a new method of nonlinear transformation （ Eigen - Mapping） to treat the eigenvalue problems in quantum mechanics as examples. The basic idea is to make the matrix of some op- erator we concerned diagonalization through introduction of small undetermined parameters and establishment equations for matrix elements. These problems can be devided into two types based on that there are or not 1 st order approximate solution for them （ in sense of the perturbation theory）. In either type we consider also two cases of degenerate and nondegenerate. To obtain solution or approximate solution for eigenvalue problems, it is given the procedures of the method and usable formulas exhaustively. Seven examples in this paper are a common sight in general course of quantum mechanics. The results from them are compared with that from oth- er approximate methods （for examples： perturbation method and variation method）. After checking the results obtained, we can see that this new method has many advantages.

关 键 词：非线性变换 本征值 近似解 

分 类 号：O413.2[理学—理论物理]