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European Journal of Physics

A physically motivated derivation of the Laplacian in terms of the total angular momentum operator

Jacob Katriel

Published Apr 5, 2019• 1 citation

Abstract

Abstract The straightforward transformation of the three-dimensional Laplacian and of the total angular momentum operator from Cartesian to spherical polar coordinates is tedious, unrewarding, and prone to errors. An alternative derivation, starting from the total angular momentum operator in Cartesian coordinates and using the generator of homogeneous scaling, readily yields the expression for the three-dimensional Laplacian in spherical polar coordinates in a manner that accounts very explicitly for the role of the total angular momentum operator. The two-dimensional case is presented as a preparatory exercise, and the N -dimensional case, which may be useful in a graduate course on the few-body problem, is shown to be as straightforward as the three-dimensional one.

DOI: 10.1088/1361-6404/ab16a1

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For pedagogical purposes it is demonstrated that with little effort the customary discussion of rotations in introductory courses can be extended to naturally produce a number of angular momentum operator properties, if coordinate-free rotation operators are used in their exponential form. This form gives rise to a particularly useful coordinate-free representation of the generators of rotations in three-space, from which the student can derive the usual angular momentum formalism for the case j=1 by elementary vector algebra alone. Since in this approach the j=1 formalism appears as a set of relations among simple geometric objects in ordinary space, it admits of a geometric interpretation and will leave the student better prepared for a more profound appreciation of the abstract concepts of the full theory.

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In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in 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 proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.