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Physical Review A

q-deformed Morse oscillator

Ian L. Cooper, Raj K. Gupta

Published Aug 1, 1995• 30 citations
Publisher

Abstract

Using the exact realization of the Morse oscillator problem in terms of SO(2,1) symmetry and its bosonization, the corresponding q deformation is performed. The ground-state energy is found to remain constant for all arbitrary (zero, real, imaginary, or complex) values of the q-deformation parameter. By fitting the resulting q-deformed energy-level spectrum to data for the hydrogen molecule in its ground electronic state, it is shown that, depending upon the value of the scaling constant, a considerable improvement results for the real and/or imaginary q deformation over the corresponding result for the classical Morse oscillator, and that the best fit is obtained in the q-deformed case with a complex value of the deformation parameter.

DOI
10.1103/physreva.52.941

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Using the exact realization of the Morse oscillator problem in terms of SO(2,1) symmetry and its bosonization, the corresponding q deformation is performed. The ground-state energy is found to remain constant for all arbitrary (zero, real, imaginary, or complex) values of the q-deformation parameter. By fitting the resulting q-deformed energy-level spectrum to data for the hydrogen molecule in its ground electronic state, it is shown that, depending upon the value of the scaling constant, a considerable improvement results for the real and/or imaginary q deformation over the corresponding result for the classical Morse oscillator, and that the best fit is obtained in the q-deformed case with a complex value of the deformation parameter.
q-deformed harmonic oscillator is established for any arbitrary complex q-deformation parameter (with real and imaginary q as its special cases) and the formalism so developed is used to construct the quantum group SUq(2). The eigenvalues of the Casimir of SUq(2) are now real, even for complex q-deformation (q=ea+ib). This happens for (a,b)≪1. Thus, the use of real part of energies in the earlier works of Gupta and collaborators7–14 is mathematically rigorous, since only small values of (a, b) are involved in all the nuclear and molecular physics problems studied7–14,18–20 so far.
q-deformation of the U(1) dynamical symmetry of U(2) group is introduced for the first time for vibrations of diatomic molecules. The question of pure real and pure imaginary value of q-deformation parameter, often raised in the literature, is found to be related to the particular choice of the constants of the model. The complex q-deformation parameter is allowed, with pure real and pure imaginary values as its special cases. Compared to the other dynamical symmetry Oq(2) of Uq(2), the result of Uq(1) of Uq(2) remains better even when the vibron number in Oq(2) is varied. The role of the vibron number in the two dynamical symmetries of U(2) is shown to be different. In Uq(1) symmetry, the vibron number is fixed by the number of observed states of the system since Uq(1) represents the q-deformation of a truncated anharmonic oscillator. A possible physical meaning to the q-deformation parameter is suggested in terms of higher order anharmonicities of the truncated anharmonic oscillator.
The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system.
Exact eigenvalues and eigenfunctions of the Schrodinger equation with generalized q-deformed Morse potential for diatomic molecules have been derived in the framework of the asymptotic iteration method (AIM). The obtained solutions have been applied to calculate the energies of bound vibrational levels of the lithium molecule in the   state. Compared to AIM previous calculations, our results are extremely accurate and are in excellent agreement with those using different approximation techniques.   Key words: q-Deformed Morse potential, asymptotic iteration method.