An approach to introduce the expression of the momentum operator in the coordinate representation
In quantum mechanics,operators and state vectors have different mathematical representations in dif-ferent representations,allowing for the choice of a suitable representation to study specific problems.In most univer-sity 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 representa-tion,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.
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