Abstract: 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.

Key words: momentum operator, coordinate representation, Taylor's translation, commutation relation