Diatomic Molecules with the Improved Deformed Generalized Deng–Fan Potential Plus Deformed Eckart Potential Model through the Solutions of the Modified Klein–Gordon and Schrödinger Equations within NCQM Symmetries
Keywords:Klein–Gordon equation, Schr¨odinger equation, deformed generalized Deng–Fan potential, deformed Eckart potential, diatomic molecules, noncommutative geometry, Bopp’s shift method, star products
In this study, the deformed Klein–Gordon equation and Schr¨odinger equations were solved with the improved deformed generalized Deng–Fan potential plus the deformed Eckart potential (IDGDFDE-P, in short) model using Bopp’s shift and standard perturbation theory methods in the symmetries of extended quantum mechanics. By employing the improved approximation to the centrifugal term, the relativistic and nonrelativistic bound-state energies are obtained for some selected diatomic molecules such as N2, I2, HCl, CH, LiH, and CO. The relativistic energy shift ΔEtotdfe (n, a, c, b, V0, V1, V2, Θ, σ, χ, j, l, s, m) and the perturbative nonrelativistic corrections ΔEnrdfe (n, α, c, b, V0, V1, V2, Θ, σ, χ, j, l, s, m) appeared as functions of the parameters (α, c, b, V0, V1, V2) and the parameters of noncommutativity (Θ, σ, χ), in addition to the atomic quantum numbers (n, j, l, s, m). In both relativistic and nonrelativistic problems, we show that the corrections to the energy spectrum are smaller than for the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present results under the IDGDFDE-P model is are consistent with what is obtained in the literature. In the new symmetries of noncommutative quantum mechanics (NCQM), it is not possible to get the exact analytical solutions for l = 0 and l ̸ = 0. Only the approximate ones can be obtained. We have clearly shown that the Schr¨odinger and Klein–Gordon equations in the new symmetries can physically describe two Dirac equations and the Duffin–Kemmer equation within the IDGDFDE-P model in the extended symmetries.
P.M. Morse. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34, 57 (1929).
H. Yukawa. On the interaction of elementary particles. I. Proc. Phys. Math. Soc. Japan 17, 48 (1935).
N. Hatami, M.R. Setare. Analytical solutions of the Klein-Gordon equation for Manning-Rosen potential with centrifugal term through Nikiforov-Uvarov method. Ind. J. Phys. 91 (10), 1229 (2017).
C. Berkdemir, A. Berkdemir, R. Sever. Editorial note: Polynomial solutions of the Schr¨odinger equation for the generalized Woods-Saxon potential. Phys. Rev. C 72, 027001 (2005)]; Phys. Rev. C 74 (3) (2006).
A. Arda, R. Sever. Bound state of the Klien-Gordon equation for Wood-Saxon potential dependent mass. Int. J. Mod. Phys. C, 19 (05), 763 (2008).
M.G. Mirand, G.H. Sun, S.H. DONG. The solution of the second P¨oschl-Tellerlike potential by Nikiforov-Uvarov method. Int. J. Mod. Phys. E 19 (01), 123 (2010).
S.M. Ikhdair, B.J. Falaye. Approximate analytical solutions to relativistic and nonrelativistic P¨oschl-Teller potential with its thermodynamic properties. Chem. Phys. 421, 84 (2013).
S.H. Dong. Relativistic treatment of spinless particles subject to a rotating Deng-Fan oscillator. Commun. Theor. Phys. 55 (6), 969 (2011).
S.M. Ikhdair, R. Sever. Any l -state improved quasi-exact analytical solutions of the spatially dependent mass Klein-Gordon equation for the scalar and vector Hulth'en potentials. Phys. Scr. 79 (3), 035002 (2009).
S.M. Ikhdair, C. Berkdemir, R. Sever. Spin and pseudospin symmetry along with orbital dependency of the Dirac-Hult'en problem. Appl. Math. Comput. 217 (22), 9019 (2011).
Z.H. Deng, Y.P. Fan. A potential function of diatomic molecules. Shandong Univ. J. 7, 162 (1957).
N.A. Hatami, J. Naji, M. Pananeh. Analytical solutions of the Klein-Gordon equation for the deformed generalized Deng-Fan potential plus deformed Eckart potential. The European Phys. J. Plus 134 (3), (2019).
G.F. Wei, C.Y. Long, X.Y. Duan, S.H. Dong. Arbitrary l wave scattering state solutions of the Schr¨odinger equation for the Eckart potential. Phys. Scr. 77 (3), 35001 (2008).
F. Cooper, A. Khare, U. Sukhatme. Supersymmetry and quantum mechanics. Physics Reports 251 (5-6), 267 (1995).
J.J. Weiss. Mechanism of proton transfer in acid-base reactions. J. Chem. Phys. 41 (4), 1120 (1964).
A. Cimas, M. Aschi, C. Barrientos, V. Ray'on, J. Sordo, A. Largo. Computational study on the kinetics of the reaction of N(4S) with CH2F. Chem. Phys. Lett. 374 (5-6), 594 (2003).
C. Eckart. The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), 1303 (1930).
W.C. Qiang, J.Y. Wu, S.H. Dong. The Eckart-like potential studied by a new approximate scheme to the centrifugal term. Phys. Scr. 79 (6), 65011 (2009).
J.J. Pe˜na, J. Garc'ıa-Mart'ınez, J. Garc'ıa-Ravelo, J. Morales. l -State solutions of multiparameter exponential-type potentials. J. Phys.: Conf. Series 490, 012199 (2014).
F. Taskin, G. Kocak. Approximate solutions of Schr¨odinger equation for Eckart potential with centrifugal term. Chin. Phys. B 19 (9), 090314 (2010).
S.H. Dong, W.C. Qiang, G.H. Sun, V.B. Bezerra. Analytical approximations to the l-wave solutions of the Schr¨odinger equation with the Eckart potential. J. Phys. A: Math. Theor. 40 (34), 10535 (2007).
J. Gao, M.C. Zhang. Analytical solutions to the Ddimensional Schr¨odinger equation with the Eckart potential. Chi. Phys. Lett. 33 (1), 010303 (2016).
B. Falaye. Any l -state solutions of the Eckart potential via asymptotic iteration method. Open Physics 10(4), 960 (2012).
X. Zou, L.Z. Yi, C.S. Jia. Bound states of the Dirac equation with vector and scalar Eckart potentials. Phys. Lett. A 346 (1-3), 54 (2005).
W.A. Yahya, K.J. Oyewumi, C.O. Akoshile, T.T. Ibrahim. Bound states of the relativistic Dirac equation with equal scalar and vector Eckart potentials using the Nikiforov-Uvarov method. Journal of Vectorial Relativity JVR 5 (3), 1 (2010).
A. Soylu, O. Bayrak, I. Boztosun. k state solutions of the Dirac equation for the Eckart potential with pseudospin and spin symmetry. J. Phys. A Math. Theor. 41 (6), 065308 (2008).
E. Ol˘gar, R. Koc, H. Tutunculer. Bound states of the s-wave Klein-Gordon equation with equal scalar and vector Standard Eckart Potential. Chin. Phys. Lett. 23 (3), 539 (2006).
Y. Zhang. Approximate analytical solutions of the KleinGordon equation with scalar and vector Eckart potentials. Phys. Scr. 78 (1), 015006 (2008).
I.O. Akpan, A.D. Antia, A.N. Ikot. Bound-state solutions of the Klein-Gordon equation with q-deformed equal scalar and vector Eckart potential using a newly improved approximation scheme. ISRN High Energy Physics 2012 Article ID 798209, 13 (2012).
A.N. Ikot, U.S. Okorie, P.O. Amadi, C.O. Edet, G.J. Rampho, R. Sever. The Nikiforov-Uvarov-Functional analysis (NUFA) method: A new approach for solving exponentialtype potentials. Few-Body Syst. 62, 9 (2021).
O.A. Awogaa, A.N. Ikotaand, J.B. Emah. Bound state solutions of deformed generalized Deng-Fan potential plus deformed Eckart potential in D-dimensions. Revista Mexicana de Fisica 59, 229 (2013).
G.F. Wei, Z.Z. Zhen, S.H. Dong. The relativistic bound and scattering states of the Manning-Rosen potential with an
improved new approximate scheme to the centrifugal term. Open Physics, Cent. Eur. J. Phys. 7 (1), 175 (2009).
C.O. Edet, U.S. Okorie, G. Osobonye, A.N. Ikot, G.J. Rampho, R. Sever. Thermal properties of Deng-Fan-Eckart potential model using Poisson summation approach. J. Math. Chem. 58, 989 (2020).
S.H. Dong, G.H. Sun, M. Lozada-Cassou. Exact solutions and ladder operators for a new anharmonic oscillator. Phys. Lett. A 340 (1-4), 94 (2005).
S.H. Dong, M. Lozada-Cassou. Exact solutions of the Klein-Gordon equation with scalar and vector ring-shaped potentials. Phys. Scr. 74 (2), 285 (2006).
D. Nath, A.K. Roy. Ro-vibrational energy analysis of Manning-Rosen and P¨oschl-Teller potentials with a new improved approximation in the centrifugal term. Eur. Phys. J. Plus 136, 430 (2021).
S. Doplicher, K. Fredenhagen, J.E. Roberts. Spacetime quantization induced by classical gravity. Phys. Lett. B 331 (1-2), 39 (1994).
E. Witten. Reflections on the fate of spacetime. Physics Today 49 (4), 24 (1996).
A. Kempf, G. Mangano, R.B. Mann. Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52 (2), 1108 (1995).
F. Scardigli. Some heuristic semi-classical derivations of the Planck length, the Hawking effect and the Unruh effect. Il Nuovo Cimento B Series 110, 1029 (1995).
R.J. Adler, D.I. Santigo. On gravity and the uncertainty principale. Mod. Phys. Lett. A 14 (20), 1371 (1999).
T. Kanazawa, G. Lambiase, G. Vilasi, A. Yoshioka. Noncommutative Schwarzschild geometry and generalized uncertainty principle. Eur. Phys. J. C 79 (2), 95 (2019).
F. Scardigli. Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment. Phys. Lett. B 452 (1-2), 39 (1999).
P.M. Ho, H.C. Kao. Noncommutative quantum mechanics from noncommutative quantum field theory. Phys. Rev. Lett. 88(15), 151602-1 (2002).
P. Gnatenko. Parameters of noncommutativity in Liealgebraic noncommutative space. Phys. Rev. D 99(2), 026009-1 (2019).
O. Bertolami, J.G. Rosa, C.M.L. De Aragao, P. Castorina, D. Zappala. Scaling of variables and the relation between no ncommutative parameters in noncommutative quantum mechanics. Mod. Phys. Lett. A 21 (10), 795 (2006).
A. Connes, M.R. Douglas, A. Schwarz. Noncommutative geometry and Matrix theory. JHEP 1998 (02), 003 (1998).
A. Maireche. A model of modified Klein-Gordon equation with modified scalar-vector Yukawa potential. Afr. Rev Phys. 15, 1 (2020).
A. Maireche. Nonrelativistic treatment of hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative quantum mechanics. Int. J. Geo. Met. Mod. Phys. 17 (5), 2050067 (2020).
A. Maireche. A recent setudy of excited energy levels of diatomics for modified more general exponential screened Coulomb potential: Extended Quantum Mechanics. J. Nano-Electron. Phys. 9 (3), 03021 (2017).
A. Maireche. Investigations on the relativistic interactions in one-electron atoms with modified Yukawa potential for spin 1/2 particles. Int. Front. Sci. Lett. 11, 29 (2017).
A. Maireche. Modified unequal mixture scalar vector Hulth'en-Yukawa potentials model as a quark-antiquark interaction and neutral atoms via relativistic treatment using the approximation of the centrifugal Term and Bopp's shift method. Few-Body Syst. 61, 30 (2020).
A. Maireche. A new approach to the approximate analytic solution of the three-dimensional Schr¨odinger equation for
hydrogenic and neutral atoms in the generalized Hellmann potential model. Ukr. J. Phys. 65 (11), 987 (2020).
A. Maireche. Effects of two-dimensional noncommutative theories on bound states Schr¨odinger diatomic molecules
under new modified Kratzer-type interactions. International Letters of Chemistry, Physics and Astronomy. 76, 1 (2017).
A. Maireche. Any L-states solutions of the modified Schr¨odinger equation with generalized Hellmann-Kratzer potential model in the symmetries of NRNCQM. To Phys. J. 4, 16 (2019).
J. Gamboa, M. Loewe, J.C. Rojas. Noncommutative quantum mechanics. Phys. Rev. D 64, 067901 (2001).
E.F. Djema¨ı, H. Smail. On quantum mechanics on noncommutative quantum phase space. Commun. Theor. Phys. (Beijing, China). 41 (6), 837 (2004).
Y. Yuan, K. Li, J.H. Wang, C.Y. Chen. Spin-1/2 relativistic particle in a magnetic field in NC phase space. Chin. Phys. C 34 (5), 543 (2010).
O. Bertolami, P. Leal. Aspects of phase-space noncommutative quantum mechanics. Phys. Lett. B 750, 6 (2015).
O. Bertolami, J.G. Rosa, C.M.L. de Aragao, P. Castorina, D. Zappala. Noncomm utative gravitational quantum well. Phys. Rev. D 72 (2), 025010-1 (2005).
J. Zhang. Fractional angular momentum in non-commutative spaces. Phys. Lett. B 584 (1-2), 204 (2004).
M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu. Hydrogen atom spectrum and the lamb shift in noncommutative QED. Phys. Rev. Lett. 86 (13), 2716 (2001).
A. Maireche. Nonrelativistic bound state solutions of the modified 2D-Killingbeck potential involving 2DKillingbeck potential and some central terms for hydrogenic atoms and quarkonium system. J. Nano-Electron. Phys. 11 (4), 04024 (2019).
A. Maireche. New bound state energies for spherical quantum dots in presence of a confining potential model at Nano and Plank's scales. NanoWorld J. 1 (4), 122 (2016).
M.A. De Andrade, C. Neves. Noncommutative mapping from the symplectic formalism. J. Mat. Phys. 59 (1), 012105 (2018).
E.M.C. Abreu, C. Neves, W. Oliveira. Noncommutative from the symplectic point of view. Int. J. Mode. Phys. A 21 (26), 5359 (2006).
E.M.C. Abreu, J.N. Ananias, A.C.R. Mendes, C. Neves, W. Oliveira, M.V. Marcial. Lagrangian formulation for noncommutative nonlinear systems. Int. J. Mod. Phys. A 27 (09), 1250053 (2012).
L. Mezincescu. Star operation in quantum mechanics. eprint. arXiv : hep-th/0007046v2.
J. Wang, K. Li. The HMW effect in noncommutative quantum mechanics. J. Phys. A: Mat. Theor. 40 (9), 2197 (2007).
K. Li, J. Wang. The topological AC effect on non-commutative phase space. Europ. Phys. J. C 50 (4), 1007 (2007).
A. Maireche. A theoretical investigation of nonrelativistic bound state solution at finite temperature using the sum of modified Cornell plus inverse quadratic potential. Sri Lankan J. Phys. 21, 11 (2020).
A. Maireche. Extended of the Schr¨odinger equation with new Coulomb potentials plus linear and harmonic radial terms in the symmetries of noncommutative quantum mechanics. J. Nano- Electron. Phys. 10 (6), 06015-1 (2018).
A. Maireche. The relativistic and nonrelativistic solutions for the modified unequal mixture of swcalar and time-like
vector Cornell potentials in the symmetries of noncommutative quantum mechanics. Jordan J. Phys. 14 (1), 59 (2021).
A. Maireche. The Klein-Gordon equation with modified Coulomb plus inverse-square potential in the noncommutative three-dimensional space. Modern Phys. Lett. A 35 (5), 052050015 (2020).
A. Maireche. A new model for describing heavy-light mesons in the extended nonrelativistic quark model under a new modified potential containing Cornell, Gaussian and inverse square terms in the symmetries Of NCQM. To Phys. J. 3, 186 (2019).
A. Maireche. Bound-state solutions of Klein-Gordon and Schr¨odinger equations for arbitrary l -state with linear combination of Hulth'en and Kratzer potentials. Afr. Rev Phys. 15, 19 (2020).
H. Motavalli, A.R. Akbarieh. Klein-Gordon equation for the Coulomb potential in noncommutative space. Mod. Phys. Lett. A 25 29), 2523 (2010).
M. Darroodi, H. Mehraban, H. Hassanabadi. The Klein Gordon equation with the Kratzer potential in the noncommutative space. Mod. Phys. Lett. A 33 (35), 1850203 (2018).
A. Saidi, M.B. Sedra. Spin-one (1 + 3)-dimensional DKP equation with modified Kratzer potential in the noncommutative space. Mod. Phys. Lett. A 35 (5), 2050014 (2019).
A. Maireche. Solutions of Klein-Gordon equation for the modified central complex potential in the symmetries of noncommutative quantum mechanics. Sri Lankan Journal of Physics. 22 (1), 1 (2021).
A. Maireche. Theoretical Investigation of the Modified Screened cosine Kratzer potential via Relativistic and Nonrelativistic treatment in the NCQM symmetries. Lat. Am. J. Phys. Educ. 14 (3), 3310-1 (2020).
A. Maireche. A theoretical model of deformed Klein-Gordon equation with generalized modified screened Coulomb plus inversely quadratic Yukawa potential in RNCQM symmetries. Few-Body Syst. 62, 12 (2021).
L. Gouba. A comparative review of four formulations of noncommutative quantum mechanics. Int. J. Mod. Phys. A 31 (19), 1630025 (2016).
F. Bopp. La m'ecanique quantique est-elle une m'ecanique statistique classique particuli'ere, Ann. Inst. Henri Poincar'e 15, 81 (1956).
M. Badawi, N. Bessis, G. Bessis. On the introduction of the rotation-vibration coupling in diatomic molecules and the factorization method. J. Phys. B: At. Mole. Phys. 5 (8), L157 (1972).
S.H. Dong, W.C. Qiang, G.H. Sun, V.B. Bezerra. Analytical approximations to the l -wave solutions of the Schr¨odinger equation with the Eckart potential. J. Phys. A: Math. Theor. 40 (34), 10535 (2007).
Y. Zhang. Approximate analytical solutions of the KleinGordon equation with scalar and vector Eckart potentials. Phys. Scr. 78 (1), 015006 (2008).
K. Bencheikh, S. Medjedel, G. Vignale. Current reversals in rapidly rotating ultracold Fermi gases. Phys. Lett. A 89 (6), 063620 (2014).
S. Medjedel, K. Bencheikh. Exact analytical results for density profile in Fourier space and elastic scattering function of a rotating harmonically confined ultra-cold Fermi gas. Phys. Lett. A 383 (16), 1915 (2019).
K.P. Gnatenko, V.M. Tkachuk. Upper bound on the momentum scale in noncommutative phase space of canonical type. EPL (Europhysics Letters). 127 (2), 20008 (2019).
K.P. Gnatenko and V.M. Tkachuk. Composite system in rotationally invariant noncommutative phase space. Int. J. Mod. Phys. A 33 (07), 1850037 (2018).
K.P. Gnatenko. Composite system in noncommutative space and the equivalence principle. Phys. Lett. A 377 (43), 3061(2013).
A. Maireche. Bound state solutions of the modified KleinGordon and S chr¨odinger equations for arbitrary l -state with the modified Morse potential in the symmetries of noncommutative quantum mechanics. J. Phys. Stud. 25 (1), 1002 (2021).
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