  We prove the existence of a ground state of the Maxwell--Schr\"odinger
equations in one spatial dimension, describing a specified amount of free
charge under the influence of a fixed charge. For one case (equal free and
fixed charge, i.e., a neutral atom), we introduce a new type of quartic Banach
space, in which the Hamiltonian is naturally coercive. We also show that for a
point charge, for any ratio of charge such that the ground state exists, it is
symmetric and decreasing.
