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Journal of Mathematical Physics

q-Deformation of the Coulomb problem

F. L. Chan, R. J. Finkelstein

Published Jul 1, 1994• 19 citations
Publisher

Abstract

The hydrogenic Schrödinger equation may be transformed into an integral equation on the group space of SO(3). The solutions of this integral equation are the Wigner functions Djmn, the matrix elements of the irreducible representations of SO(3). In this article we examine a q-deformation of the Coulomb problem for which the new wave functions are the matrix elements of the irreducible representations of the quantum group SUq(2). The new integral equation is formulated in terms of the Woronowicz integral. The eigenvalues are given by a modified Balmer formula which lifts the Coulomb degeneracy and replaces n by [n].

DOI
10.1063/1.530466

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Physical Review A•Jul 1, 1995•5 citations
The hydrogenic Schrödinger equation may be transformed into an integral equation on the group space of SO(3). The solutions of this integral equation are the Wigner functions Djmn, the matrix elements of the irreducible representations of SO(3). In this article we examine a q-deformation of the Coulomb problem for which the new wave functions are the matrix elements of the irreducible representations of the quantum group SUq(2). The new integral equation is formulated in terms of the Woronowicz integral. The eigenvalues are given by a modified Balmer formula which lifts the Coulomb degeneracy and replaces n by [n].
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
A deformation of quantum mechanics obtained by replacing commutators by q-commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q-harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra.
The general problem of describing quantum states not only by wave functions and density operators but also by probability distribution functions is discussed in this paper. The qubit-state density-matrix elements expressed in terms of probability distributions are connected with the irreducible representation of group SU(2). The transform of the probability distributions corresponding to the unitary transform of the qubit state of the spin-1/2 system is obtained and expressed in terms of matrix elements of the group SU(2). The possibility to extend the introduced formalism to other quantum states is suggested. For qubit systems, the notion of density operators algebra and the relation with the probability representation of quantum states are discussed. The Schrödinger equation for qubit states is obtained as an equation for probabilities.
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.