  We investigate non-relativistic quantum mechanical potentials between
fermions generated by various classes of QFT operators and evaluate their
singularity structure. These potentials can be generated either by four-fermion
operators or by the exchange of a scalar or vector mediator coupled via
renormalizable or non-renormalizable operators. In the non-relativistic regime,
solving the Schr\"odinger equation with these potentials provides an accurate
description of the scattering process. This procedure requires providing a set
of boundary conditions. We first recapitulate the procedure for setting the
boundary conditions by matching the first Born approximation in quantum
mechanics to the tree-level QFT approximation. Using this procedure, we show
that the potentials are non-singular, despite the presence of terms
proportional to $r^{-3}$ and $\nabla_{i}\nabla_{j}\delta^{3}(\vec{r})$. This
surprising feature leads us to propose the \emph{Quantum Mechanics Swampland},
in which the Landscape consists of non-relativistic quantum mechanical
potentials that can be UV completed to a QFT, and the Swampland consists of
pathological potentials which cannot. We identify preliminary criteria for
distinguishing potentials which reside in the Landscape from those that reside
in the Swampland. We also consider extensions to potentials in higher
dimensions and find that Coulomb potentials are non-singular in an arbitrary
number of spacetime dimensions.
