Approximate Bound State Solutions of the Hellmann Plus Kratzer Potential in N-dimensional Space

Year 2020, , 791 - 804, 01.09.2020

Aysel Özfidan

https://doi.org/10.35378/gujs.672684

Cited By: 2

Abstract

We have examined the approximate l_(N-1)-state solutions of the N-dimensional Schrödinger equation for a particle interacting with the Hellmann plus Kratzer potential. In hyperspherical coordinate system, we have constructed the bound state energy equation and the wavefunctions expressed by the hypergeometric function via the asymptotic iteration approach in detail. When considered the special cases of parameters in Hellmann plus Kratzer potential, this potential turns into several potential models. In this connection, the non-relativistic energy spectra for the modified Kratzer, Yukawa, Coulomb and Hellmann potentials in approximate analytic form have been obtained in hyperspherical coordinates. We have presented the numerical energy eigenvalues for the Hellmann, Yukawa and Coulomb potentials in N=3 dimensions. Our present results provide an appropriate test of the accuracy of asymptotic iteration formalism.

Keywords

Hellmann, Kratzer, N-dimensions

References

• [1] Greene, R.L., Aldrich, C., “Variational wave functions for a screened Coulomb potential”, Physical Review A, 14 (6): 2363-2366, (1976).
• [2] Ciftci, H., Hall, R.L., Saad, N., “Asymptotic iteration method for eigenvalue problems”, Journal of Physics A: Mathematical and General, 36(47): 11807-11816, (2003).
• [3] Ciftci, H., Hall, R.L., Saad, N., “Construction of exact solutions to eigenvalue problems by the asymptotic iteration method”, Journal of Physics A: Mathematical and General, 38 (5): 1147-1155, (2005).
• [4] Ciftci, H., Hall, R.L., Saad, N., “Iterative solutions to the Dirac equation”, Physical Review A, 72 (2): 022101-7, (2005).
• [5] Louck, J.D., Shaffer, W.H., “Generalized orbital angular momentum and the n-fold degenerate quantum mechanical oscillator: Part I the twofold degenerate oscilator”, Journal of Molecular Spectroscopy, 4 (1-6): 285-297, (1960).
• [6] Louck, J.D,“Generalized orbital angular momentum and the n-fold degenerate quantum mechanical oscillator : Part II the n-fold degenerate oscillator” Journal of Molecular Spectroscopy, 4 (1-6): 298-333, (1960).
• [7] Louck, J.D,“Generalized orbital angular momentum and the n-fold degenerate quantum mechanical oscillator : Part III radial integrals” Journal of Molecular Spectroscopy, 4 (1-6): 334-341, (1960).
• [8] Chatterjee, A., “Large-N expansions in quantum mechanics, atomic physics and some O(N) invariant systems”, Physics Reports, 186 (6): 249-370, (1990).
• [9] Hellmann, H.,“ A new approximation method in the problem of many electrons”, The Journal of Chemical Physics, 3 (1): 61, (1935).
• [10] Hellmann, H., Kassatotchkin, W., “Metallic Binding According to the combined approximation procedure”, The Journal of Chemical Physics, 4(5): 324-325, (1935).
• [11] Kratzer, A.,“ Die ultraroten rotationsspektren der halogenwasserstoffe”, Zeitschrift für Physik, 3 (5): 289-307, (1920).
• [12] Dutt, R., Mukherji, U., Varshni, Y.P.,“ An improved calculation for screened Coulomb potentials in Rayleigh-Schrodinger perturbation theory”, Journal of Physics A: Mathematical and General, 18: 1379-1388, (1985).
• [13] Vrscay, E.R.,“ Hydrogen atom with a Yukawa potential: Perturbation theory and continued-fractions–Padé approximants at large order”, Physical Review A, 33(2): 1433-1436, (1986).
• [14] Stubbins, C.,“ Bound states of the Hulthén and Yukawa potentials”, Physical Review A, 48(1): 220-227, (1993).
• [15] Adamowski, J.,“ Bound eigenstates for the superposition of the Coulomb and the Yukawa potentials”, Physical Review A, 31(1): 43-50, (1985).
• [16] Hamzavi, M., Thylwe, K.E, Rajabi, A.A.,“ Approximate bound states solutions of the Hellmann potential”, Communications in Theoretical Physics, 60(1): 1-8, (2013).
• [17] Simons, G., Parr, R.G., Finlan, J.M.,“ New alternative to the Dunham potential for diatomic molecules”, The Journal of Chemical Physics, 59(6): 3229-3234, (1973).
• [18] Molski, M, Konarski, J.,“ Extended Simons-Parr-Finlan approach to the analytical calculation of the rotational-vibrational energy of diatomic molecules”, Physical Review A, 47(1): 711-714, (1993).
• [19] Pliva, J.,“A closed rovibrational energy formula based on a modified Kratzer potential”, Journal of Molecular Spectroscopy, 193(1): 7-14, (1999).
• [20] Oyewumi, K.J., “Realization of the spectrum generating algebra for the generalized Kratzer potentials”, International Journal of Theoretical Physics, 49: 1302, (2010).
• [21] Edet, C.O., Okorie, K.O., Louis, H., Nzeata-Ibe, N.,“ Any l-state solutions of the Schrödinger equation interacting with Hellmann-Kratzer potential model”, Indian Journal of Physics, (2019).
• [22] Dong, S.H., Sun, G.H., “The Schrödinger equation with a Coulomb plus inverse-square potential in D-dimensions”, Physica Scripta, 70(2-3): 94-97, (2004).
• [23] Durmus, A., “Nonrelativistic treatment of diatomic molecules interacting with a generalized Kratzer potential in hyperspherical coordinates”, Journal of Physics A: Mathematical and General, 44(15): 155205-14, (2011).
• [24] Ikdhair, S.M., Sever, R., “Exact solutions of the pseudo-Coulomb potential plus ring-shaped potential in the D-dimensional Schrödinger equation by the Nikiforov-Uvarov method”, arXiv: quany-ph/0703042v1 2007; 1-15.
• [25] Gradshteyn, I.S., Ryzhik, I.M., “Tables of Integrals, Series and Products”. Academic Press, New York, (2000).
• [26] Ebomwonyi, O., Onate, C.A., Onyeaju, M.C., Ikot, A.N., “Any l-state solutions of the Schrödinger equation interacting with Hellmann-generalized Morse potential model”, Karbala International Journal of Modern Science, 3(1): 59-68, (2017).