分离变量法在量子力学中的应用

Application of Variable Separation Method in Quantum Mechanics

分离变量法是求解偏微分方程常用的方法之一,其核心思想是将一个涉及多个变量的偏微分方程化成几个偏微分方程或者常微分方程分别求解,使问题得到简化,最终得以解决.在求解薛定方程、氢原子体系、自由粒子狄拉克方程等方面,分离变量法起到了化整为零、化难为易的作用.梳理与总结的解题思路有益于量子力学课程初学者的理解与提升.

The variable separation method is one of common methods to solve partial differential equations. Its core idea is to transform a partial differential equation involving multiple variables into several partial differential equations or ordinary differential equations to solve separately, so that the problem can be simplified and finally solved. In solving the Schrodinger equation, the hydrogen atom system and the free particle Dirac equation, the variable separation method has played a role in breaking the whole into parts and making it easier. The idea of sorting and summarizing is beneficial to the understanding and improvement of beginners in quantum mechanics learning.