The $S{O}_q(N,\mathbf{R})$-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ${\mathbf{R}}_q^N$ and its Hilbert Space Structure

Abstract

We show that the isotropic harmonic oscillator in the ordinary euclidean space ${\mathbf{R}}^N$ ($N\ge 3$) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups), $S{O}_q(N,\mathbf{R})$. The q-deformation is the consequence of replacing $R^N$ by ${\mathbf{R}}_q^N$ (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over ${\mathbf{R}}_q^N$, which we use for the definition of the scalar product of states.

55 pages, tex, to appear in the Int. J Mod. Phys. A, 1993. Revised Feb. 1993

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