  For one-dimensional many-body systems interacting via the \textit{Coulomb
force} and with \textit{arbitrary} external potential energy, we derive
(\textit{i}) the \textit{node coalescence condition} for the wave function.
This condition rigorously proves the following: (\textit{ii}) that the
particles satisfy \textit{only} a node coalescence condition; (\textit{iii})
that irrespective of their charge or statistics, the particles cannot coalesce;
(\textit{iv}) that the particles cannot cross each other, and must be ordered;
(\textit{v}) the particles are therefore distinguishable; (\textit{vi}) as such
their statistics are not significant; (\textit{vii}) conclusions similar to
those of the spin-statistics theorem of quantum field theory are arrived at via
non-relativistic quantum mechanics; (\textit{viii}) the noninteracting system
\textit{cannot} be employed as the lowest-order in a perturbation theory of the
interacting system. (\textit{ix}) Finally, the coalescence condition for
particles with the short-ranged delta-function interaction and
\textit{arbitrary} external potential energy, is also derived. These particles
can coalesce and cross each other.
