Interaction of Spinless Particles with Yukawa Ring-Shaped Potential

  • A. D. Antia Theoretical Physics Group, Department of Physics, Faculty of Science, University of Uyo
  • E. E. Ituen Theoretical Physics Group, Department of Physics, Faculty of Science, University of Uyo
  • U. S. Jim Department of Mathematics, Faculty of Science, University of Uyo
  • E. E. Eyibio Theoretical Physics Group, Department of Physics, Faculty of Science, University of Uyo
Keywords: spinless particles, Yukawa potential, angle-dependent potential, approximation scheme, Nikiforov–Uvarov method

Abstract

We have obtained the approximate solutions of the Klein–Gordon equation with the Yukawa ring-shaped potential, by using the Nikiforov–Uvarov method for a special case of equal scalar and vector potentials. The energy eigenvalues for bound states and the corresponding wave functions are also obtained in a proper approximation. We have also shown that the results can be used to evaluate the energy eigenvalues of the Yukawa, angle-dependent, and Coulomb potentials. The numerical results are discussed and presented in the table and in the figure, which suggest their applicability to other systems. With the adjusted potential parameters given in the table, it is shown that the interaction of spinless (Klein–Gordon) particles with the Yukawa ring-shaped potential gives positive energy eigenvalues for the various quantum states.

References

X.A. Zhang, K. Chen, Z.L. Duan. Bound state of Klein–Gordon equation and Dirac equation for ring-shaped nonspherical oscillator scalar and vector potentials. Chin. Phys. 14, 42 (2005).

https://doi.org/10.1088/1009-1963/14/1/009

S.H. Dong, G.H. Sun, M. Lozada-Cassou. Exact solutions and ladder operators for a new anharmonic oscillator. Phys. Lett. A 340, 94 (2005).

https://doi.org/10.1016/j.physleta.2005.04.024

C.Y. Chen, S.H. Dong. Exactly complete solutions of the Coulomb potential plus a new ring-shaped potential. Phys. Lett. A 335, 374 (2005).

https://doi.org/10.1016/j.physleta.2004.12.062

A.D. Alhaidari. Scattering and bound states for a class of non-central potential. J. Phys. A: Math. Gen. 38, 3409 (2005).

https://doi.org/10.1088/0305-4470/38/15/012

M. Kibler, L.G. Mardoyan, G.S. Pogosyan. On a generalized Kepler–Coulomb system: interbasis expansions. Int. J. Quantum Chem. 52, 1301 (1994).

https://doi.org/10.1002/qua.560520606

W. Gereiner, Relativistic Quantum Mechanics, Wave Equations (Springer, 2000).

https://doi.org/10.1007/978-3-662-04275-5

C. Quense. Supersymmetry and the Dirac oscillator. Int. J. Mod. Phys. A 6, 1567 (1991).

https://doi.org/10.1142/S0217751X91000836

F. Ya¸suk, C. Berkdemir, A. Berkdemir. Exact solutions of the Schr¨odiger equation with non-central potential. J. Phys. A: Math. Gen. 38, 6579 (2005).

https://doi.org/10.1088/0305-4470/38/29/012

Y. Xu, S. He, C.S. Jia. Approximate analytical solutions of the Dirac equation with the Poschl-Teller potential including spin-orbit coupling. J. Phys. A: Math. Theor. 42, 198002 (2009).

https://doi.org/10.1088/1751-8113/42/19/198002

X.Y. Liu, G. F.Wei, C.Y. Long. Arbitrary wave relativistic bound state solutions for the Eckart potential. Int. J. Theor. Phys. 48, 463 (2009).

https://doi.org/10.1007/s10773-008-9821-z

T. Chen, Y.F. Diao, C.S. Jia. Bound state solutions of the Klein–Gordon equation with the generalized Poschl–Teller potential. Phys. Scr. 79, 065014 (2009).

https://doi.org/10.1088/0031-8949/79/06/065014

B. Sutherland. Exact coherent states of a one dimensional quantum fluid in a time-dependent trapping potential. Phys. Rev. Lett. 80, 3678 (1998).

https://doi.org/10.1103/PhysRevLett.80.3678

C.F. Hou, Y. Li, Z.X. Zhou. Bound states of the Klein–Gordon equation and Dirac equation with scalar and vector Morse-type potentials. Acta Phys. Sinica 48, 1999 (1999).

A. Soylu, O. Bayrak, I. Boztosun. An approximate solution of Dirac–Hulthen problem with pseudospin and spin symmetry for any state. J. Math. Phys. 48, 082302 (2007).

https://doi.org/10.1063/1.2768436

A. Soylu, O. Bayrak, I. Boztosun. K-state solution of the Dirac equation for the Eckart potential with the spinorbit coupling term and spin symmetry. J. Phys. A: Math. Theor. 41, 065308 (2008).

https://doi.org/10.1088/1751-8113/41/6/065308

M. Hamzavi, H. Hassanabadi, A.A. Rajabi. Exact solutions of the Dirac equation with Hartmann potential. Int. J. Mod. Phys. E 19, 2189 (2010).

https://doi.org/10.1142/S0218301310016594

J.Y. Guo. Bound states of relativistic particle in tan ()- type potential. Acta Phys. Sin. 51, 1453 (2002).

M. C. Zhang, Z.B. Wang. Exact solutions of the Klein–Gordon equation with a new anharmonic oscillatior potential. Chin. Phys. Lett. 22, 2994 (2005).

https://doi.org/10.1088/0256-307X/22/12/003

C.S. Jia, P. Guo, X.L. Peng. Exact solution of the Dirac–Eckart problem with spin and pseudospin symmetry. J. Phys. A: Math. Gen. 39, 7737 (2006).

https://doi.org/10.1088/0305-4470/39/24/010

W.C. Qiang, R.S. Zhou, Y. Gao. Application of the exact quantization rule to the relativistic solution of the rotational Morse potential with pseudospin symmetry. J. Phys. A: Math. Theor. 40, 1677 (2007).

https://doi.org/10.1088/1751-8113/40/7/016

A. Arda, R. Sever, C. Tezcan. Analytical solutions to the Klein–Gordon equation with position-dependent mass for −parameter Poschl–Teller potential. Chinese J. Phys. 48, 27 (2010).

https://doi.org/10.1088/0256-307X/27/1/010306

S.H. Dong, X.Y. Gu. Arbitrary −state solutions of the Schr¨odiger equation with the Deng-Fan molecular potential. J. Phys.: Conf. Ser. 96, 012109 (2008).

https://doi.org/10.1088/1742-6596/96/1/012109

O. Aydogdu, R. Sever. Solutions of the Dirac equation for pseudoharmonic potential by using Nikiforov–Uvarov method. Phys. Scr. 80, 015001 (2009).

https://doi.org/10.1088/0031-8949/80/01/015001

S.M. Ikhdair. Approximate solutions of the Dirac equation for the Rosen–Morse potential including the spin-orbit centrifugal term. J. Math. Phys. 51, 023525 (2010).

https://doi.org/10.1063/1.3293759

M. Hamzavi, A.A. Rajabi, H. Hassanabadi. Exact spin and pseudospin symmetry solutions of the Dirac equation for Mie-type potential including a Coulomb-like tensor potential. Few-Body Syst. 48, 171 (2010).

https://doi.org/10.1007/s00601-010-0095-7

C. Berkdemir, R. Sever. Pseudospin symmetry solution of the Dirac equation with an angle-dependent potential. J. Phys. A: Math. Theor. 41, 045302 (2008).

https://doi.org/10.1088/1751-8113/41/4/045302

M. Hamzavi, H. Hassanabadi, A.A. Rajabi. Exact solution of Dirac equation for Mie-type potential using the Nikiforov–Uvarov method under the pseudospin and spin symmetry limit. Mod. Phys. Lett. A 25, 2447 (2010).

https://doi.org/10.1142/S0217732310033402

G.H. Sun, S.H. Dong. New type shift operators for threedimensional infinite well potential. Mod. Phys. Lett. A 26, 351(2011).

https://doi.org/10.1142/S0217732311034815

G.F. Wei, S.H. Dong. Approximately analytical solutions of the Manning–Rosen potential with the spin-orbit coupling term and spin symmetry. Phys. Lett. A 373, 49(2008).

https://doi.org/10.1016/j.physleta.2008.10.064

N. Kandirmaz, R. Sever. Coherent states for PT-non-PTsymmetric and non-Hermitian Morse potential via the path integral method. Phys. Scr. 81, 035302(2010).

https://doi.org/10.1088/0031-8949/81/03/035302

H. Xian-Quan, L. Guang, W. Zhi-Min, N. Lian-Bin, M. Yan. Solving Dirac equation with new ring-shaped nonspherical harmonic oscillator potential. Commun. Theor. Phys. 53, 242 (2010).

https://doi.org/10.1088/0253-6102/53/2/07

F. Ya¸suk, A. Durmus, I. Boztosun. Exact analytical solutions of the relativistic Klein–Gordon equation with noncentral equal scalar and vector potential. J. Math. Phys. 47, 082302 (2006).

https://doi.org/10.1063/1.2227258

M.C. Zhang, G.H. Sun, S.H. Dong. Exactly complete solutions of the Schr¨odiger equation with a spherically Harmonic oscillator ring-shaped potential. Phys. Lett. A 374, 704 (2010).

https://doi.org/10.1016/j.physleta.2009.11.072

O. Bayrak, M. Karakoc, I. Boztosun, R. Sever. Approximate analytical solution of the Schr¨odiger equation for Makarov potential with any angular momentum. Int. J. Theor. Phys. 47, 3005 (2008).

https://doi.org/10.1007/s10773-008-9735-9

C.Y. Chen, C.L. Liu, F.L. Lu. Exact solutions of Schr¨odiger equation for the Makarov potential. Phys. Lett. A 374, 1346 (2010).

https://doi.org/10.1016/j.physleta.2010.01.018

H. Yukawa. Interaction of elementary particle. Proc. Phys. Math. Soc. Japan 17, 48 (1935).

J. McEnnan, L. Kissel, R.H. Pratt. Analytic perturbation theory for screened Coulomb potentials: non-relativistic case. Phys. Rev. A 13, 532 (1976).

https://doi.org/10.1103/PhysRevA.13.532

A.D. Antia, S.E. Etuk, A.O. Adeniran. Solutions of relativistic Klein–Gordon equation with equal scalar and vector shifted Hulthen plus angle dependent potential. Adv. Phys. Theor. Appl. 47, 45 (2015).

Y. Xu, S. He, C.S. Jia. Approximate analytical solutions of the Klein–Gordon equation with the Poschl–Teller potential including the centrifugal term. Phys. Scripta 81, 045001 (2010).

https://doi.org/10.1088/0031-8949/81/04/045001

A.S. Davydov. Quantum Mechanics (Pergamon Press, 1976) [ISBN: 0-08-020437-6].

A.D. Antia, E.E. Umo, C.C. Umoren. Solutions of nonrelativistic Schr¨odiger equation with Hulthen–Yukawa plus angle-dependent potential within the frame work of Nikiforov–Uvarov method. J. Theor. Phys. and Cryp. 10, 1 (2015).

A. Alhaidari, H. Bahlouli, I. Nasser, M. Abdelmonem. An efficient mapped pseudospectral method for weakly bound state. J. Chem. Phys. A 41, 032001 (2008).

S.M. Ikhdair, R. Sever. Approximate analytical solutions of the general Wood–Saxon potential including spin-orbit coupling term and spin symmetry. Central European Journal of Physics 5, 2322 (2010).

A.D. Antia, O.P. Akpan. Yukawa-angle dependent potential and its applications to diatonic molecules under Schr¨odiger wave equation. J. Appl. Theor. Phys. Res. 1, 9 (2017).

https://doi.org/10.24218/jatpr.2017.08

A.F. Nikiforov, V.B. Uvarov. Special Functions of Mathematical Physics (Birkh¨auser, 1988).

https://doi.org/10.1007/978-1-4757-1595-8

C. Tezcan, R. Sever. A general approach of the exact solution of the Schr¨odiger equation. Int. J. Theor. Phys. 48, 339 (2009).

https://doi.org/10.1007/s10773-008-9806-y

M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Washington, 1998).

I.S. Gradshein, I.M. Rhyzhik. Tables of Integrals, Series and Products (Elsevier, 2007).

C. Berkdemir, Y.F. Chen. The exact solutions of the Dirac equation with a novel angle- dependent potential. Phys. Scr. 79, 035003 (2009).

https://doi.org/10.1088/0031-8949/79/03/035003

Majid Hamzavi, A.A. Rajabi. Exact solutions of the Dirac equation with Coulomb plus a novel angle-dependent potential. Z. Naturforsch. 66a, 533 (2011).

R.L. Greene, C. Aldrich. Variational wave functions for a screened Coulomb potential. Phys. Rev. A 14, 2363 (1976).

https://doi.org/10.1103/PhysRevA.14.2363

O.A. Awoga, A.N. Ikot, 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).

A.N. Ikot, A.D. Antia, I.O. Akpan, O.A. Awoga. Bound state solutions of Schr¨odinger equation with modified Hylleraas plus exponential Rosen–Morse potential. Revista Mexicana de Fisica 59, 46 (2013).

Published
2018-12-12
How to Cite
Antia, A., Ituen, E., Jim, U., & Eyibio, E. (2018). Interaction of Spinless Particles with Yukawa Ring-Shaped Potential. Ukrainian Journal of Physics, 62(10), 913. https://doi.org/10.15407/ujpe62.10.0913
Section
General problems of theoretical physics