  Solvability of the ubiquitous quantum harmonic oscillator relies on a
spectrum generating osp(1|2) superconformal symmetry. We study the problem of
constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a
given space of states. This problem stems from interacting higher spin models
coupled to gravity. In one dimension, we show that the solution to this problem
is the Plyushchay family of quantum mechanical models with hidden
superconformal symmetry obtained by viewing the harmonic oscillator as a one
dimensional Dirac system, so that Grassmann parity equals wavefunction parity.
These models--both oscillator and particle-like--realize all possible unitary
irreducible representations of osp(1|2).
