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Journal of Physics A Mathematical and General

q-deformed ladder and shift operators for three exactly solvable potentials obeying SO(2,1) symmetry

R K Gupta, Ian L. Cooper

Published Dec 7, 1994• 4 citations
Publisher

Abstract

The quantum deformation algebra SO q (2,1) is studied and applied to derive the q-analogues of the ladder and shift operators for the radial Coulomb, radial harmonic oscillator and Morse oscillator potentials. The q-deformed operators in all three cases are found to act like shift operators, called q-shift operators. Their possible similarity with the quasi-shift operators arising in supersymmetric quantum mechanics, or factorization, of the radial harmonic oscillator is also pointed out.

DOI
10.1088/0305-4470/27/23/024

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The quantum deformation algebra SO q (2,1) is studied and applied to derive the q-analogues of the ladder and shift operators for the radial Coulomb, radial harmonic oscillator and Morse oscillator potentials. The q-deformed operators in all three cases are found to act like shift operators, called q-shift operators. Their possible similarity with the quasi-shift operators arising in supersymmetric quantum mechanics, or factorization, of the radial harmonic oscillator is also pointed out.
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.
Morse oscillator is one of the known solvable potentials which attracts many applications in quantum mechanics especially in quantum chemistry. One of the interesting results of this study is the generation of ladder operators for Morse potential. The operators are a representation of the shifting energy levels of the states exhibited by the wavefunction. From this result, we manipulate and deform the operators in such a way that it gives a noncommutative property to promote noncommutative quantum mechanics (NCQM). The resultant NC feature can be shown in the spatial coordinates and finally the Hamiltonian. In this study, we consider two-dimensional Morse potential where the ladder operators are in the form of the corresponding 2D Morse.
Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e., the spectrum of one can be obtained from another (with possible exception of the lowest level) by the q2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra su q(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane.
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.