简并情况下两个对易算符的共同本征函数系的简单求解方法

算符理论是量子力学课程的重要教学内容,其中两个力学量算符的对易及其本征函数问题是算符理论中的一个难点。简并情况下两个对易算符的本征函数系问题更为复杂,很多传统教材中都没有进行详细分析和讨论。该文利用一种简单的方法证明了简并情况下两个对易算符具有共同本征函数系的结论,并提出了寻找两个对易算符的共同本征函数系的两种简单方法—交集法与线性组合法。该文有助于教师和学生深刻地领会和求解两个对易算符的共同本征函数问题。

Simple Methods for Solving Simultanious Eigenfunctions of Two Commutative Operators in Degenerate Situation

The operator theory is a very important part in the quantum mechanics, in which two mechanical quantity operator commutative problem and their simultanious eigenfunctions are difficult points in the operator theory as they are more complex especially in degenerate situation. Moreover, detailed analysis and discussion is not given in many traditional textbooks. In this paper, a simple proof that two commutative operators can hold the simultanious eigenfunctions in degenerate situation is pres- ented, and we propose two simple methods to solve simultanious eigenfunetions of two commutative operators in degenerate sit uation： the intersection method and the linear combination method. This can help teachers and students deeply understand and solve the problems of the simultanious eigenfunctions of two commutative operators.