Nonlinear Dynamics

, Volume 92, Issue 2, pp 659–671 | Cite as

Novel quantum description of time-dependent molecular interactions obeying a generalized non-central potential

  • Salah Menouar
  • Jeong Ryeol Choi
  • Ramazan Sever
Original Paper


Quantum features of time-dependent molecular interactions are investigated by introducing a time-varying Hamiltonian that involves a generalized non-central potential. According to the Lewis–Riesenfeld theory, quantum states (wave functions) of such dynamical systems are represented in terms of the eigenstates of the invariant operator. Hence, we have derived the eigenstates of the invariant operator of the system using elegant mathematical manipulations known as the unitary transformation method and the Nikiforov–Uvarov method. Based on full wave functions that are evaluated by considering such eigenstates, quantum properties of the system are analyzed. The time behavior of probability densities which are the absolute square of the wave functions are illustrated in detail. This research provides a novel method for investigating quantum characteristics of complicated molecular interactions. The merit of this research compared to conventional ones in this field is that time-varying parameters, necessary for the actual description of intricate atomic and molecular behaviors with high accuracy, are explicitly considered.


Time-dependent systems Non-central potential Modified Kratzer potential Unitary transformation Schrödinger equation 



This research was supported by the Basic Science Research Program of the year 2017 through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No.: NRF-2016R1D1A1A09919503).


  1. 1.
    Arulsamy, A.D.: Quantum adiabatic theorem for chemical reactions and systems with time-dependent orthogonalization. Prog. Theor. Phys. 126(4), 577–595 (2011)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kosloff, R.: Time-dependent quantum-mechanical methods for molecular dynamics. J. Phys. Chem 92(8), 2087–2100 (1988)CrossRefGoogle Scholar
  3. 3.
    Baer, M.: Time-dependent molecular fields created by the interaction of an external electromagnetic field with a molecular system. Int. J. Quantum Chem. 114(24), 1645–1659 (2014)CrossRefGoogle Scholar
  4. 4.
    Jadhav, S., Konstantopoulos, K.: Fluid shear- and time-dependent modulation of molecular interactions between PMNs and colon carcinomas. Am. J. Physiol. Cell Physiol. 283(4), C1133–C1143 (2002)CrossRefGoogle Scholar
  5. 5.
    McLachlan, A.D., Gregory, R.D., Ball, M.A.: Molecular interactions by the time-dependent Hartree method. Mol. Phys. 7(2), 119–129 (1964)CrossRefGoogle Scholar
  6. 6.
    Cai, J.-P., Harris, B., Falanga, V., Eaglstein, W.H., Mertz, P.M., Chin, Y.-H.: Recruitment of mononuclear cells by endothelial cell binding into wound skin is a selective, time-dependent process with define molecular interactions. J. Invest. Dermatol. 95(4), 415–421 (1990)CrossRefGoogle Scholar
  7. 7.
    Stehlik, D., Brunner, H., Hausser, K.H.: Time-dependent interactions in di-tert-butyl-nitroxide (DBNO) as studied by ESR and proton relaxation. J. Mol. Struct. 1(1), 25–30 (1967)CrossRefGoogle Scholar
  8. 8.
    Runge, K., Micha, D.A., Feng, E.Q.A.: Time-dependent molecular orbital approach to electron transfer in ion-atom collsions. Int. J. Quantum Chem. 38(S24), 781–790 (1990)CrossRefGoogle Scholar
  9. 9.
    Lewis Jr., H.R.: Classical and quantum systems with time-dependent harmonic-oscillator-type hamiltonians. Phys. Rev. Lett. 18(15), 510–512 (1967)CrossRefGoogle Scholar
  10. 10.
    Lewis Jr., H.R., Riesenfeld, W.B.: An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. J. Math. Phys. 10(8), 1458–1473 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ermakov, V.P.: Second-order differential equations. Conditions of complete integrability. Univ. Isz. Kiev Series III 9, 1–25 (1880) (translation by A.O. Harin). See: Appl. Anal. Discrete Math. 2(2), 123–148 (2008)Google Scholar
  12. 12.
    Dodonov, V.V., Man’ko, V.I., Rosa, L.: Quantum singular oscillator as a model of a two-ion trap: an amplification of transition probabilities due to small-time variations of the binding potential. Phys. Rev. A 57(4), 2851–2858 (1998)CrossRefGoogle Scholar
  13. 13.
    Menouar, S., Maamache, M., Saadi, Y., Choi, J.R.: Exact wavefunctions for a time-dependent Coulomb potential. J. Phys. A Math. Theor. 41(21), 215303 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Menouar, S., Maamache, M., Choi, J.R., Sever, R.: On the quantization of one-dimensional nonstationary Coulomb potential system. J. Phys. Soc. Jpn. 81(6), 064003 (2012)CrossRefGoogle Scholar
  15. 15.
    Dodonov, V.V., Malkin, I.A., Manko, V.I.: Even and odd coherent states and excitations of a singular oscillator. Physica 72(3), 597–615 (1974)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Choi, J.R., Gweon, B.H.: Operator method for a nonconservative harmonic oscillator with and without singular perturbation. Int. J. Mod. Phys. B 16(31), 4733–4742 (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ndong, M., Tal-Ezer, H., Kosloff, R., Koch, C.P.: A Chebychev propagator with iterative time ordering for explicitly time-dependent Hamiltonians. J. Chem. Phys. 132(6), 064105 (2010)CrossRefGoogle Scholar
  18. 18.
    Agboola, D., Oladipupo, A.: Complete analytic solutions of the Mie-type potentials in N-dimensions. Acta Phys. Polonica A 120(3), 371–377 (2011)CrossRefGoogle Scholar
  19. 19.
    Falaye, B.J., Oyewumi, K.J., Sadikoglu, F., Hamzavi, M., Ikhdair, S.M.: Analysis of quantum-mechanical states of the ring-shaped Mie-type diatomic molecular model via the Fisher’s information. J. Theor. Comput. Chem. 14(05), 1550036 (2015)CrossRefGoogle Scholar
  20. 20.
    Blado, G.G.: Supersymmetry and the Hartmann potential of theoretical chemistry. Theor. Chim. Acta 94(1), 53–66 (1996)CrossRefGoogle Scholar
  21. 21.
    Chen, C.-Y., Liu, C.-L., Lu, F.-L.: Exact solutions of Schrödinger equation for the Makarov potential. Phys. Lett. A 374(11–12), 1346–1349 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Aygun, M., Bayrak, O., Boztosun, I., Sahin, Y.: The energy eigenvalues of the Kratzer potential in the presence of a magnetic field. Eur. Phys. J. D 66(2), 1 (2012). (Article 35) CrossRefGoogle Scholar
  23. 23.
    Oyewumi, K.J.: Analytical solutions of the Kratzer-Fues potential in an arbitrary number of dimensions. Found. Phys. Lett. 18(1), 75–84 (2005)CrossRefGoogle Scholar
  24. 24.
    Frank, W.M., Land, D.J., Spector, R.M.: Singular potentials. Rev. Mod. Phys. 43(1), 36–96 (1971)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sadeghi, J., Pourhassan, B.: Exact solution of the non-central modified Kratzer potential plus a ring-shaped like potential by the factorization method. Electron. J. Theor. Phys. 5(17), 193–202 (2008)zbMATHGoogle Scholar
  26. 26.
    Nasser, I., Abdelmonem, M.S., Abdel-Hady, A.: Handling the singularities of the perturbed Kratzer and inverted Kratzer potentials. In: Proceedings of the 8th Conference on Nuclear and Particle Physics, pp. 123-130 (2011)Google Scholar
  27. 27.
    Setare, M.R., Karimi, E.: Algebraic approach to the Kratzer potential. Phys. Scr. 75(1), 90–93 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Berkdemir, C., Berkdemir, A., Han, J.: Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential. Chem. Phys. Lett. 417(4–6), 326–329 (2006)CrossRefGoogle Scholar
  29. 29.
    Kratzer, A.: Die ultraroten Rotationsspektren der Halogenwasserstoffe. Z. Phys. 3(5), 289–307 (1920)CrossRefGoogle Scholar
  30. 30.
    Fues, E.: Zur Intensität der Bandenlinien und des Affinitätsspektrums zweiatomiger Moleküle. Ann. Phys. (Paris) 386(19), 281–313 (1926)zbMATHGoogle Scholar
  31. 31.
    Ikhdair, S.M., Sever, R.: Relativistic treatment in D-dimensions to a spin-zero particle with noncentral equal scalar and vector ring-shaped Kratzer potential. Cent. Eur. J. Phys. 6(1), 141–152 (2008)Google Scholar
  32. 32.
    Ikhdair, S.M., Sever, R.: Polynomial solution of PT-/non-PT-symmetric and non-Hermitian generalized Woods-Saxon potential via Nikiforov–Uvarov method. Int. J. Theor. Phys. 46(6), 1643–1665 (2007)CrossRefzbMATHGoogle Scholar
  33. 33.
    Liboff, R.L.: Introductory Quantum Mechanics, 4th edn. Addison Wesley, San Fransisco (2002)zbMATHGoogle Scholar
  34. 34.
    Erdély, A.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)Google Scholar
  35. 35.
    Wei, W., Xie, Z., Cooper, L.N., Seidel, G.M., Maris, H.J.: Study of exotic ions in superfluid Helium and the possible fission of the electron wave function. J. Low Temp. Phys. 178(1), 78–117 (2015)CrossRefGoogle Scholar
  36. 36.
    Plebanski, J.: Wave functions of a harmonic oscillator. Phys. Rev. 101(6), 1825–1826 (1956)CrossRefGoogle Scholar
  37. 37.
    Raymer, M.G.: Measuring the quantum mechanical wave function. Contemp. Phys. 38(5), 343–355 (1997)CrossRefGoogle Scholar
  38. 38.
    Merad, M., Bensaid, S.: Wave functions for a Duffin-Kemmer-Petiau particle in a time-dependent potential. J. Math. Phys. 48(7), 073515 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Bengtsson, J., Lindroth, E., Selstø, S.: Wave functions associated with time-dependent, complex-scaled Hamiltonians evaluated on a complex time grid. Phys. Rev. A 85(1), 013419 (2012)CrossRefGoogle Scholar
  40. 40.
    Oh, H.G., Lee, H.R., George, T.F., Um, C.I.: Exact wave functions and coherent states of a damped driven harmonic oscillator. Phys. Rev. A 39(11), 5515–5522 (1989)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392(1802), 45–57 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ben-Aryeh, Y.: Berry and Pancharatnam topological phases of atomic and optical systems. J. Opt. B Quantum Semiclassical Opt. 6(4), R1–R18 (2004)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Unanyan, R.G., Fleischhauer, M.: A geometric phase gate without dynamical phases. Phys. Rev. A 69(5), 050302(R) (2004)CrossRefGoogle Scholar
  44. 44.
    Choi, J.R., Yeon, K.H.: Coherent states of the inverted Caldirola–Kanai oscillator with time-dependent singularities. Ann. Phys. (NY) 323(4), 812–826 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics. Birkhäuser, Basel (1998)zbMATHGoogle Scholar
  46. 46.
    Menouar, S., Choi, J.R.: Quantization of time-dependent singular potential systems in one-dimension by using the Nikiforov–Uvarov method. J. Korean Phys. Soc. 67(7), 1127–1132 (2015)CrossRefGoogle Scholar
  47. 47.
    Menouar, S., Choi, J.R.: Quantization of time-dependent non-central singular potential systems in three dimensions by using the Nikiforov–Uvarov method. J. Korean Phys. Soc. 68(4), 505–512 (2016)CrossRefGoogle Scholar
  48. 48.
    Yaşuk, F., Berkdemir, C., Berkdemir, A.: Exact solutions of the Schrödinger equation with non-central potential by the Nikiforov–Uvarov method. J. Phys. A Math. Gen. 38(29), 6579–6586 (2005)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratory of Optoelectronics and Compounds (LOC), Department of Physics, Faculty of ScienceUniversity of Ferhat Abbas Setif 1SétifAlgeria
  2. 2.Department of PhysicsKyonggi University, Gwanggyosan-ro, Yeongtong-gu, SuwonGyeonggi-doRepublic of Korea
  3. 3.Department of PhysicsMiddle East Technical UniversityAnkaraTurkey

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