  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.

Quantum Physics