Some intricacies of the momentum operator in quantum mechanics

Abstract

In quantum mechanics textbooks, the momentum operator is defined in Cartesian coordinates and the form of the momentum operator in spherical polar coordinates is rarely discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls into a trap. In this work, we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator, taking spherical polar coordinates as our specimen coordinate system and proposing an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.