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Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms

Carl W David

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The conversion between representations of angular momentum in spherical polar and cartesian form is discussed.

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Physica Scripta•Jan 1, 1993•7 citations
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.
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 spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.