Arbitrary l-solutions of the Schrödinger equation interacting with Hulthén – Hellmann potential model

E. S. William, E. P. Inyang, E. A. Thompson

Abstract


In this study, we obtained bound state solutions of the radial Schrödinger equation by the superposition of Hulthén plus Hellmann potential within the framework of Nikiforov-Uvarov (NU) method for an arbitrary  - states. The corresponding normalized wave functions expressed in terms of Jacobi polynomial for a particle exposed to this potential field was also obtained. The numerical energy eigenvalues for different quantum state have been computed. Six special cases are also considered and their energy eigenvalues are obtained. Our results are found to be in good agreement with the results in literature. The behavior of energy in the ground and excited state for different quantum state are studied graphically.


Keywords


Schrödinger equation; Nikiforov-Uvarov method; eigenvalues; eigenfunction; Hulthén -Hellmann potential

Full Text:

PDF

References


W. Greiner Quantum Mechanics, An Introduction 4th Ed. (Springer Verlag Berlin Heidelberg, New York, 2000), pp.

C. A. Cari, A. Suparmi, H. Azizah, The approximate solution of Schrodinger equation with minimal length presence for Yukawa potential, AIP Conference Proceedings, 020093, (2018), https://doi.org/10.1063/1.5054497

C. P. Onyenegecha, C. A. Onate, O. K. Echendu, A. A. Ibe, H. Hassanabadi, Solutions of Schrodinger equation for the modified Mobius square plus Kratzer potential. Eur. Phys. J. Plus, (2020) 135, doi:10.1140/epjp/s13360-020-00304-z

R. H. Parmar, K. R. Purohit, A. K. Rai, Approximaate analytical solution of the extended Hulthen – Yukawa with inverse square and Coulombic term plus ring shape potential, AIP Conference proceedings 2220 (2020)140071, https//doi.org/10.1063/5.0001432

H. I. Ahmadov, M. V. Qocayeva, N. S. Huseynova, The bound state solutions of the D-dimensional Schrödinger equation for the Hulthén potential within SUSY quantum mechanics. Int. J. Mod. Phys E 26 (2017) 1 – 18.

https://doi.org/10.1142/S0218301317500288

A. N. Ikot, U. S. Okorie, G. Osobonye, P.O. Amadi, C. O. Edet, M. J. Sithole, G. J. Rampho, R. Sever, Superstatistics of Schrodinger equation with pseudo-harmonic potential in external magnetic and Aharanov-Bohm fields, Heliyon 6 (2020) e03738, doi: 10.1016/j.heliyon.2020.e03738

R. V. Torres, G. H. Sun, S. H. Dong, Quantum information entropy for a hyperbolical potential function, Phys. Scr, 90 (2015) 035205.

G. H. Sun, M. A. Aoki, S. H. Dong, Quantum information entropies of the eigenstates

for the Poschl—Teller-like potential, Chin. Phys. B 22 (2013) 050302.

https://doi.org/10.1088/1674-1056/22/5/050302

A. I. Ahmadov, M. Naeem, M. V, Qocayeva, V. A. Tarverdiyeva, Analytical Solutions of the Schrödinger Equation for the Manning-Rosen plus Hulthén Potential Within SUSY Quantum Mechanics, J. Phys. Conf. Ser., 965 (2018) 012001, doi:10.1088/1742-6596/965/1/012001

I. Toli, S. Zou, Schrödinger equation with Coulomb potential admits no exact solutions. ‎Chem. Phys. Lett. X 2 (2019) 100021, doi:10.1016/j.cpletx.2019.100021

M. N. Oqielat, A. El-Ajou, Z. Al-Zhour, R. Alkhasawneh, H. Alrabaiah, Series solutions for nonlinear time-fractional Schrödinger equations: Comparisons between conformable and Caputo derivatives, Alex. Eng. J. (2020), doi:10.1016/j.aej.2020.01.023

T. Das, A. Arda, A. Exact Analytical Solution of theN-Dimensional Radial Schrödinger Equation with Pseudoharmonic Potential via Laplace Transform Approach. High Energy Phys (2015) 1–8. doi:10.1155/2015/137038

H. Dutta, H. Günerhan, K. K. Ali, R. Yilmazer, Exact Soliton Solutions to the Cubic-Quartic Non-linear Schrödinger Equation With Conformable Derivative, Front. Phys, 8, (2020), doi:10.3389/fphy.2020.00062

M. Abu-Shady, A. N. Ikot, Analytic solution of multi-dimensional Schrödinger equation in hot and dense QCD media using the SUSYQM method. The Eur Phys J Plus, 134 (2019) 7. doi:10.1140/epjp/i2019-12685-y

C. A. Onate, Approximate solutions of the non-relativistic schrodinger equation with an interaction of coulomb and Hulthen potentials, SOP tran. Theor. Phy., (2014) issn(print): 2372-2487 issn(online): 2372-2495,(1,2).

A. D. Antia, E. E. Umo, C. C. Umoren, Solutions of non relativistic Schrödinger equation with Hulthen-Yukawa plus angle dependent potential within the framework of Nikiforov-Uvarov method, J. Theor. Phys. Crypt, 10 (2015). 1 – 8, doi: 10.11648/j.ajpc.20150405.11

A. Antia, E. Ituen, Non-relativistic Treatment of Schrodinger particles under inversely quadratic Hellmann plus Ring-Shaped potentials, Ukr. J. Phys, 62, (2018), 7 633. https://doi.org/10.15407/ujpe62.07.0633

M. Hamzavi, S. M. Ikhdair, M. Solaimani, A semirelativistic treatment of spinless particles subject to the yukawa potential with arbitrary angular momenta, Int. J. Mod. Phys. E, 21 (2012), 1250016. Doi:10.1142/s0218301312500164

M. Hamzavi, K. E. Thylwe, A. A. Rajabi, Approximate Bound States Solution of the Hellmann Potential, Commun Theor Phys, 60 (2013) 1–8. doi:10.1088/0253-6102/60/1/01

A. K. Rai, D. P. Rathaud, The mass spectra and decay properties of dimesonic states, using the Hellmann potential, ‎Eur. Phys. J C 75 (2015) (9). doi:10.1140/epjc/s10052-015-3695-z

I. Nasser, M. S. Abdelmonem, A. Abdel – Hady, Scaling behavior of the Hellmann potential with different strength parameters, Intl. J. Inter. Chem. Phys, 112 (2014) 19, 2608 – 2613.

J. Stanek, Approximate analytical solutions for arbitrary l-state of the Hulthén potential with an improved approximation of the centrifugal term, Cent Eur J Chem, 9 (2011) 4 737-742, https://doi.org/10.2478/s11532-011-0050-6

S. M. Ikhdair, B. J. Falaye, Two Approximate Analytic Eigensolutions of the Hellmann Potential with any Arbitrary Angular Momentum, Verl. Z. Naturforsch., (2013)

B. I. Ita, A. I. Ikeuba, Solutions to the Schrödinger Equation with Inversely Quadratic Yukawa Plus Inversely Quadratic Hellmann Potential Using Nikiforov-Uvarov Method, J. Phys. B. Atom. Molec. Phys, 582610 (2013), https://doi.org/10.1155/2013/582610

B. I.Ita, Solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for any angular momentum quantum number using the Nikiforov-Uvarov method, Intl J. Rec. Adv. Phys, 4 (2013) 2, 1 – 9, https://doi.org/10.1155/2013/582610

L. Hulthén, Uber die eigenlosunger der schrodinger - gleichung des deuterons, Ark. Mat. Astron. Fys. A 28(1942) 5

S. Hassanabadi, M. Ghominejad, S. Zarrinkamar, H. Hassanabadi, The Yukawa potential in semirelativistic formulation via supersymmetry quantum mechanics, Chin. Phys. B, 22 (2013) 6

https://doi.org/10.1088/1674-1056/22/6/060303

H. Taseli, Bessel basis with applications: N-dimensional isotropic polynomial oscillators,

Int. J. Quant. Chem (1997) 935-947, DOI: 10.1002/(SICI)1097-461X(1997)63:53.0.CO;2-X

O. Bayrak, I. Boztosun, Bound state solutions of the Hulthén potential by using the asymptotic iteration method, Phys. Scri., 76 (2007) 1, 92–96. doi:10.1088/0031-8949/76/1/016.

I. B. Okon, O. Popoola, E. E. Ituen, Bound state solution to Schrodinger equation with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametricNikiforovUvarov method, Intl J. Rec. adv. Phys. ,5 (2016) 2, DOI : 10.14810/ijrap.2016.5101 1

H. Hellmann, A New Approximation Method in the Problem of Many Electrons, J Chem Phys 3(1935) 61, https://doi.org/10.1063/1.1749559

S. Ikhdair, R. Sever, Exact solutions of the radial Schrödinger equation for some physical potentials, Open Phys, 5 (2007) 4, doi:10.2478/s11534-007-0022-9

A. N. Ikot, S. E. Etuk, H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, Indian J. Phys 89 (2015) 289, https://doi:10.1007/s12648-014-0558-7

C. A. Onate, M. C. Onyeaju, A. N. Ikot, Analytical solutions of the Dirac equation under Hellmann–Frost–Musulin potential, Ann. Phys, (2016) http://dx.doi.org/10.1016/j.aop.2016.10.006

C. O. Edet, P.O. Okoi, Any l-state solutions of the Schrodinger equation for q-deformed Hulthen plus generalized inverse quadratic Yukawa potential in arbitrary dimensions, Rev Mex Fis, 65 (2019) 333-344, DOI: https://doi.org/10.31349/RevMexFis.65.333

C. A. Onate, O. Ebomwonyi, K. O. Dopamu, J. O. Okoro, M. O. Oluwayemi, Eigen solutions of the D-dimensional Schrӧdinger equation with inverse trigonometry scarf potential and Coulomb potential, Chin. J. Phys, (2018), doi:10.1016/j.cjph.2018.03.013

P.O. Okoi, C. O. Edet, T. O. Magu, Relativistic treatment of the Hellmann generalized

Morse potential, Rev. Mexic. Fisica, 66 (2020) 1, https://doi.org/10.31349/RevMexFis.66.1

B. Ita, P. Tchoua, E. Siryabe, G. E. Ntamack, Solutions of the Klein-Gordon equation with the Hulthen potential using the Frobenius method, Intl. J. Theor. Math. Phys, 4 (2014) 5, 173 – 177, doi:10.5923/j.ijtmp.20140405.02

C. O. Edet, O. Kalu, K. O. Okorie, H. Louis, N. A. Nzeata-Ibe, Any l-state solutions of the Schrodinger equation interacting with Hellmann–Kratzer potential model, Indian J. Phys, (2019) 1 – 9., https://.doi.2F10.10072Fs12648-019-01467-x

L. Hitler, B. I. Ita, P. Tchoua, A. A. Ettah, Bound State Solutions of the Klein-Gordon Equation for the More General Exponential Screened Coulomb Potential Plus Yukawa (MGESCY) Potential Using Nikiforov-Uvarov Method. J. Phys Math, 9 (2018) 1, doi:10.4172/2090-0902.1000261

O. Ebomwonyi, C. A. Onate, M. C. Onyeaju, A. N. Ikot, Any ℓ − states solutions of the Schrödinger equation interacting with Hellmann-generalized Morse potential model, Karbala Intl J. Mod. Sc, 3 (2017) 1, 59–68. doi:10.1016/j.kijoms.2017.03.001

A. F. Nikiforov, V. B. Uvarov, Special Functions of Mathematical Physics, (Birkhauser, Bassel, 1988)

C. Berkdemir, A. Berkdemir,J. Han, Bound state solutions of the Schr¨odinger equation for modified Kratzer’s molecular potential, Chem. Phys. Lett, 417 (2006) 4–6, 326-329, DOI: 10.1016/j.cplett.2005.10.039

L. I. Schiff, Quantum Mechanics, (McGraw-Hill, New York, NY, USA, 1955)

K. J. Oyewumi, O. J. Oluwadare, The scattering phase shifts of the Hulth´en-type potential plus Yukawa potential, Eur. Phys. J. Plus, 131 2016) 295

R. L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential, Phys. Rev. A, 14 (1976) 6, 2363–2366, doi:10.1103/physreva.14.2363

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, (2020b)1 – 25, https://doi.org/10.1007/s10910-020-01107-4

O. Bayrak, G. Kocak, I. Boztosun, Any l-state solutions of the Hulthén potential by the asymptotic iteration method, Journal of Phys A Math. Gen, 39 (2006) 37, 11521, https://doi.org/10.1088/0305-4470/39/37/012

C. S. Jia, J. Y. Liu, P. Q. Wang, New Approximation Scheme for the Centrifugal Term and Hulthén Potential, Phys. Lett. A, 372 (2008) 4779-4782, https://doi.org/10.1007/s10773-009-0051-9

I. B. Okon, O. Popoola, Bound- State solution of Schrodinger equation with Hulthen plus generalized exponential Coulomb potential using Nikiforov-Uvarov method, Intl. J. of Rec. adv. Phys, 4 (2015) 3, DOI : 10.14810/ijrap.2015.4301 1

S. M. Ikhdair, R. Sever, Bound Energy for the Exponential-Cosine-Screened Coulomb Potential. J. Math. Chem., 41 (2006) 4, 329–341, doi:10.1007/s10910-006-9080-2

C. A. Onate, J. O. Ojonubah, A. Adeoti, E. J. Eweh, M. Ugboja, Approximate Eigen Solutions of D.K.P. and Klein-Gordon Equations with Hellmann Potential, Afr. Rev. Phys, 9:0062 497 (2014), 1 – 8.

S. M. Ikhdair, R. Sever, A perturbative treatment for the bound states of the Hellmann potential, Journal of Molecular Structure: THEOCHEM 809 (2007) 103–113

Qiang, W.C., Gao, Y., Zhou, R.S. (2008). Arbitrary l-state approximate solutions of the Hulthén potential through the exact quantization rule, Cen. Eur. Phys. J. Phys., 6, 356, DOI: 10.2478/s11534-008-0041-1

S. M. Ikhdair, An improved approximation scheme for the centrifugal term and the Hulthén potential. The Eur. Phys. J. A, 39 (2009) 3, 307–314. doi:10.1140/epja/i2008-10715-2




DOI: https://doi.org/10.31349/RevMexFis.66.730

Refbacks



Revista Mexicana de Física

ISSN: 2683-2224 (on line), 0035-001X (print)

Bimonthly publication of Sociedad Mexicana de Física, A.C.
Departamento de Física, 2o. Piso, Facultad de Ciencias, UNAM.
Circuito Exterior s/n, Ciudad Universitaria. C. P. 04510 Ciudad de México.
Apartado Postal 70-348, Coyoacán, 04511 Ciudad de México.
Tel/Fax: (52) 55-5622-4946, (52) 55-5622-4840. rmf@ciencias.unam.mx