简谐振子波函数的代数解及Hermite多项式的递推  

Algebraic Approach to Wave Function of Harmonic Oscillator and Recursion Relations of Hermite Polynomial

作　　者：王帮美[1] 胡先权[1] 

机构地区：[1]重庆师范大学物理学与信息技术学院

出　　处：《重庆师范大学学报（自然科学版）》2009年第3期119-122,共4页Journal of Chongqing Normal University:Natural Science

基　　金：重庆市教委基础理论研究基金(No.KJ060812)

摘　　要：简谐振子模型是量子力学中极其简单又重要的模型,其物理思想在其他相关的学科中都有着广泛的应用,通过多种途径去深入理解简谐振子模型,对理解量子力学的实质和运用量子力学作为工具去研究微观物理模型都有重要的意义;另一方面在实际工作中应用代数方法去求解力学量的本征值和波函数是研究量子力学的主要手段。以简谐振子为例,运用代数方法,先给出一维简谐振子的波函数,从而推广到多维简谐振子,并结合相应算符的对易关系给出Hermite多项式及其递推关系,回避了通过级数展开去求解Hermite方程的过程;同时指出《厄米本征值问题的探究》一文中的不足之处。Harmonic oscillator model, the physical method of which enjoys wide application in other related courses is the simplest but the most important model in quantum mechanics. On one hand, profound understanding of harmonic oscillator model through various ways is crucial to understand the essence of quantum mechanics and research into micro physical models by means of quantum mechanics ; on the other hand, the application of algebraic approach to solve the eigenvalues and wave function of mechanics is the main means to solve the actual problems in quantum mechanics. This paper will take harmonic oscillator as an example, and apply algebraic approach to firstly present wave function of one-dimensional harmonic oscillator, and then multi-dimensional harmonic oscillator, and also combine corresponding commutator relations of operator to present Hermite polynomials and its recursion relations, which will avoid the process of using series to solve the equation of Heimite. Meanwhile, it will improve some shortcomings revealed in the same paper.

关 键 词：升降算符 简谐振子 波函数 HERMITE多项式 

分 类 号：O411.1[理学—理论物理]