动量算符在坐标表象中表示式的一种引入方法

An approach to introduce the expression of the momentum operator in the coordinate representation

在量子力学中,算符和态矢量在不同的表象下有不同的数学表达形式,因此可以选择合适的表象来研究特定的问题.在大学物理教材中,对动量算符在坐标表象下表达式的引入,大多都是基于自由粒子态在坐标表象下的波函数是平面波的基本假设,导致其引入过程在数学上不够严密,造成学生困惑.本文结合泰勒平移算符数学公式和量子力学基本对易关系式,另辟新径推导动量算符在坐标表象中的数学表达形式.这种引入方法在数学上更严密,且在教学上更有启发性.

In quantum mechanics,operators and state vectors have different mathematical representations in different representations,allowing for the choice of a suitable representation to study specific problems.In most university physics textbooks,the introduction of the expression of the momentum operator in the coordinate representation is based on the basic assumption that the wave function of a free particle in the coordinate representation is a plane wave,which leads to a less rigorous mathematical derivation and causes confusion among students.In this paper,we propose a new approach to derive the mathematical expression of the momentum operator in the coordinate representation,combining the mathematical formula of Taylor's translation operator and the fundamental commutation relation in quantum mechanics.This introduction method is more mathematically rigorous and more inspiring in teaching.