Quantum mechanics in the E(3 vertical bar 2) superspace

Quantum mechanics in the E(3\2) superspace is presented. The even Grassmann variables and the odd ones are treated on the same footing. The conjugate momenta and the superspin operators are realized as differential operators acting on functions defined on E(3\2). For central potential, the problem h as both OSP(3\2) and super-rotation symmetries which allows us to separate odd Grassmann variables by means of the superspin basis of the superspheric al harmonics. Then a radial equation is derived and this equation is solved in two cases: for the Coulomb potential bound states, where the energy lev els are the same as for the hydrogen atom, and for the isotropic harmonic o scillator, where they are those of the one-dimensional oscillator. In both cases the energy levels are degenerate, but the states are labelled by the eigenvalues of the superspin and of the spin. Finally the Fock; space of th e oscillator is constructed.