Symmetric and supersymmetries of the quantum harmonic oscillator

The supersymmetric version of the one-dimensional harmonic oscillator is studied by taking into account its conformal properties. The largest superalgebra of symmetries and supersymmetries is derived as Osp(2/2)per square Sh(1), the semidirect sum of Osp(2/2) and the Heisenberg superalgebra. Through a one-to-one correspondence between the non-relativistic free case and the harmonic oscillator description, the (expected) supersymmetries of the Schroedinger equation are deduced. The above structure appears as the largest spectrum-generating superalgebra of the harmonic oscillator and its representation within an energy basis is given. The physical three-dimensional case is also considered when the maximal set of (super)symmetries is required and this case is compared with recent work. (author)