利用诺特定理和泰勒展开引入薛定谔方程

Schrodinger equation introduced by using Noether's theorem and Taylor's expansion

在量子力学中,薛定谔方程用于描述微观粒子运动状态随时间变化的规律,其重要意义不言而喻.在传统教学中薛定谔方程一般作为定义直接被引入或者由实验事实波粒二象性出发逐步导引建立从而被引入,但传统的导引建立方法具有一定跳跃性,逻辑不严谨,缺乏深层原理支撑,不利于部分学生的深入理解.本文旨在从对称性和守恒量存在一一对应关系的深层原理架构出发,以泰勒展开为基础进行数学推导,在学生已经具备一定量子力学和数学知识的基础上循序渐进地引入薛定谔方程.首先,文中介绍了传统的引入方法.其次,在回顾泰勒展开的基础上引入了泰勒平移的概念,形成新旧知识的有机结合并进一步激发学生创造性思维.最后,利用泰勒平移概念结合诺特定理自然引出了薛定谔方程.

In quantum mechanics,Schrdinger equation is used to describe how the motion of microscopic particles changes with time,and its significance is self-evident.In traditional teaching,Schrdinger equation is usually introduced directly as a definition,or gradually guided and established by the wave-particle duality of experimental facts.However,the traditional guidance and construction methods have certain leaps,their logic is not rigorous,and they lack deep-seated principle support,which is not conducive to the in-depth understanding of some students.The purpose of this paper is to start from the deep-seated theoretical framework of symmetry and one-to-one correspondence of conservation quantities,carries out mathematical derivation on the basis of Taylor's expansion,and gradually introduces Schrdinger equation on the basis of students'quantum mechanics and mathematical knowledge.Firstly,we introduce the traditional introduction methods.Secondly,on the basis of reviewing Taylor's expansion,we introduce Taylor's translation concept,which forms an organic combination of old and new knowledge and further stimulates students'creative thinking.Finally,the Schrdinger equation is naturally derived by using the concept of Taylor's translation and Noether's theorem.