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Quantum mechanics is often developed in the position representation, but thisis not necessary, and one can perform calculations in arepresentation-independent fashion, even for wavefunctions. In this work, weillustrate how one can determine wavefunctions, aside from normalization, usingonly operators and how those operators act on state vectors. To do this inplane polar and spherical coordinates requires one to convert the translationoperator into those coordinates. As examples of this approach, we illustratethe solution of the Coulomb problem in two and three dimensions without needingto express any operators in position space.

journal of mathematical physics

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