  The hydrogen atom is a system amenable to an exact treatment within
Schroedinger's formulation of quantum mechanics according to coordinates in
four systems -- spherical polar, paraboloidal, ellipsoidal and spheroconical
coordinates; the latter solution is reported for the first time. Applications
of these solutions include angular momenta, a quantitative calculation of the
absorption spectrum and accurate plots of surfaces of amplitude functions. The
shape of an amplitude function, and even the quantum numbers in a particular
set to specify such an individual function, depend on the coordinates in a
particular chosen system, and are therefore artefacts of that particular
coordinate representation within wave mechanics. All discussion of atomic or
molecular properties based on such shapes or quantum numbers therefore lacks
general significance
