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arXiv:hep-th/9507116 · Journal of Mathematical Physics

A q-deformation of the Coulomb problem

James Feigenbaum, Peter G. O. Freund

Published Apr 1, 1996• 13 citations
arXivHTMLPDF

Abstract

The algebra of observables of SOq(3)-symmetric quantum mechanics is extended to include the inverse 1/R of the radial coordinate and used to obtain eigenvalues and eigenfunctions of a q-deformed Coulomb Hamiltonian.
DOI
10.1063/1.531471

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Transparent closed forms of the radial SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) Schr\"odinger equation acting on the noncommutative quantum Euclidean space are written both for l=0 and l\ensuremath{\ne}0. Starting from the s-wave equation leads to the derivation of an alternative q analog of the radial Schr\"odinger equation in N space dimensions exhibiting a modified symmetry, now in terms of wave functions depending on the radial coordinate only. The harmonic oscillator and the Coulomb system are treated as concrete examples. These results open the way to the derivation of q-deformed (1/N)-energy formulas for arbitrary spherically symmetrical potentials. Comparisons with exact SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) results, as well as further interpretations, have also been made.
The observable algebra O of SO_q(3)-symmetric quantum mechanics is generated by the coordinates of momentum and position spaces (which are both isomorphic to the SO_q(3)-covariant real quantum space R_q^3). Their interrelations are determined with the quantum group covariant differential calculus. For a quantum mechanical representation of O on a Hilbert space essential self- adjointness of specified observables and compatibility of the covariance of the observable algebra with the action of the unitary continuous corepresent- ation operator of the compact quantum matrix group SO_q(3) are required. It is shown that each such quantum mechanical representation extends uniquely to a self-adjoint representation of O. All these self-adjoint representations are constructed. As an example an SO_q(3)-invariant Coulomb potential is intro- duced, the corresponding Hamiltonian proved to be essentially self-adjoint and its negative eigenvalues calculated with the help of a q-deformed Lenz-vector.
A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.
We found hermitian realizations of the position vector r⃗\vec{r}r, angular momentum Λ⃗\vecΛΛ and linear momentum p⃗\vec{p}p​ behaving like vectors with respect to the SUq(2)SU_q(2)SUq​(2) algebra, generated by L0L_0L0​ and L±L_\pmL±​. They are used to write the qqq deformed Schrödinger equation, whose solution for Coulomb and oscillator potential are briefly discussed.