  We study the quantum mechanical harmonic oscillator in two and three
dimensions, with particular attention to the solutions as represents of their
respective symmetry groups: O(2), O(3), and O(2,1). Solving the Schrodinger
equation by separating variables in polar coordinates, we obtain wavefunctions
characterized by a principal quantum number, the group Casimir eigenvalue, and
one observable component of orbital angular momentum, with eigenvalue $m+s$,
for integer $m$ and real constant parameter $s$. In each symmetry group, $s$
splits the solutions into two inequivalent representations, one associated with
$s=0$, which recovers the familiar description of the oscillator as a product
of one-dimensional solutions, and the other with $s>0$ (in three dimensions,
$s=0, 1/2$) whose solutions are non-separable in Cartesian coordinates, and are
hence overlooked by the standard Fock space approach. In two dimensions, a
single set of creation and annihilation operators forms a ladder representation
for the allowed oscillator states for any $s$, and the degeneracy of energy
states is always finite. However, in three dimensions, the integer and
half-integer eigenstates are qualitatively different: the former can be
expressed as finite dimensional irreducible tensors under O(3) or O(2,1), and a
ladder representation can be constructed via irreducible tensor products of the
vector creation operator multiplet, while the latter exhibit infinite
degeneracy and the finite-dimensional ladder representation fails for these
states. These results are closely connected to the breaking of a unitary
symmetry of the harmonic oscillator Hamiltonian recently discussed by Bars.
