  The three-dimensional potential equation, motivated by representations of
quantum mechanics, is investigated in four different scenarios: (i) In the
usual Euclidean space $\mathbb{E}_{3}$ where the potential is singular but
invariant under the continuous inhomogeneous orthogonal group
$\mathcal{I}O(3)$. The invariance under the translation subgroup is compared to
the corresponding unitary transformation in the Schr\"{o}dinger representation
of quantum mechanics. This scenario is well known but serves as a reference
point for the other scenarios. (ii) Next, the discrete potential equation as a
partial difference equation in a three-dimensional lattice space is studied. In
this arena the potential is non-singular but invariance under $\mathcal{I}O(3)$
is broken. This is the usual picture of lattice theories and numerical
approximations. (iii) Next we study the six-dimensional continuous phase space.
Here a phase space representation of quantum mechanics is utilized. The
resulting potential is singular but possesses invariance under
$\mathcal{I}O(3)$. (iv) Finally, the potential is derived from the discrete
phase space representation of quantum mechanics, which is shown to be an
\emph{exact} representation of quantum mechanics. The potential function here
is both non-singular and possesses invariance under $\mathcal{I}O(3)$, and this
is proved via the unitary transformations of quantum mechanics in this
representation.
