Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion

TL;DR

This paper demonstrates a method to determine quantum wavefunctions in a representation-independent way by converting translation operators into plane polar and spherical coordinates, simplifying calculations like the Coulomb problem.

Contribution

It introduces a novel approach to quantum mechanics calculations that avoids position representation by converting translation operators into coordinate-specific forms.

Findings

Successfully solved Coulomb problem in 2D and 3D without position space operators

Showed how to determine wavefunctions using only operators and their actions

Provided a coordinate transformation method for translation operators

Abstract

Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.