从诺特定理引入对易关系的新教学法

A new teaching method of commutation relation from Noether s theorem

量子力学中算符对易关系对于学生理解量子力学具有至关重要的意义,也与海森伯测不准原理直接相关.在本科阶段的课程中,算符对易关系公式通常是理论上直接给出的,或者其导出过程依赖于选定的具体表象下算符的具体表示形式,算符对易关系公式的引入过程没有与经典物理产生有机的联系,缺乏普遍性容易被学生误解.基于经典和量子物理都普遍成立的诺特定理物理体系守恒量与对称性相对应的想法,本文旨在引入一个更具启发性的算符对易关系导出教学方法,为学生提供一个简单而自然的理解思路.

The commutation relation of operators in quantum mechanics is of great significance for students to understand quantum mechanics,and is also directly related to Heisenberg s uncertainty principle.In undergraduate courses,the commutation relation of operators is usually directly given in theory,or its derivation depends on the specific representation of operators under the selected concrete representation.The introduction process of the commutation relation formula of operators has no organic connection with classical physics,and the lack of universality is easy to be misunderstood by students.The idea is based on the universally established Noether s theorem in both classical and quatum physics,which corresponds to the consevation of quantities and symmetries in the physical system.This paper aims to introduce a more enlightening teaching method for the derivation of operator commutation relations,providing students with a simple and natural way of understanding.