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

- 55 Downloads

## Abstract

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.

## Keywords

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

### Acknowledgements

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).

## References

- 1.Arulsamy, A.D.: Quantum adiabatic theorem for chemical reactions and systems with time-dependent orthogonalization. Prog. Theor. Phys.
**126**(4), 577–595 (2011)CrossRefMATHGoogle Scholar - 2.Kosloff, R.: Time-dependent quantum-mechanical methods for molecular dynamics. J. Phys. Chem
**92**(8), 2087–2100 (1988)CrossRefGoogle Scholar - 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.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.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.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.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.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.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.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)MathSciNetCrossRefMATHGoogle Scholar - 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.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.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)MathSciNetCrossRefMATHGoogle Scholar - 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.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.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)CrossRefMATHGoogle Scholar - 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.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.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.Blado, G.G.: Supersymmetry and the Hartmann potential of theoretical chemistry. Theor. Chim. Acta
**94**(1), 53–66 (1996)CrossRefGoogle Scholar - 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)MathSciNetCrossRefMATHGoogle Scholar - 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.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.Frank, W.M., Land, D.J., Spector, R.M.: Singular potentials. Rev. Mod. Phys.
**43**(1), 36–96 (1971)MathSciNetCrossRefGoogle Scholar - 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)MATHGoogle Scholar - 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.Setare, M.R., Karimi, E.: Algebraic approach to the Kratzer potential. Phys. Scr.
**75**(1), 90–93 (2007)MathSciNetCrossRefMATHGoogle Scholar - 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.Kratzer, A.: Die ultraroten Rotationsspektren der Halogenwasserstoffe. Z. Phys.
**3**(5), 289–307 (1920)CrossRefGoogle Scholar - 30.Fues, E.: Zur Intensität der Bandenlinien und des Affinitätsspektrums zweiatomiger Moleküle. Ann. Phys. (Paris)
**386**(19), 281–313 (1926)MATHGoogle Scholar - 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.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)CrossRefMATHGoogle Scholar - 33.Liboff, R.L.: Introductory Quantum Mechanics, 4th edn. Addison Wesley, San Fransisco (2002)MATHGoogle Scholar
- 34.Erdély, A.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)Google Scholar
- 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.Plebanski, J.: Wave functions of a harmonic oscillator. Phys. Rev.
**101**(6), 1825–1826 (1956)CrossRefGoogle Scholar - 37.Raymer, M.G.: Measuring the quantum mechanical wave function. Contemp. Phys.
**38**(5), 343–355 (1997)CrossRefGoogle Scholar - 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)MathSciNetCrossRefMATHGoogle Scholar - 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.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.Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A
**392**(1802), 45–57 (1984)MathSciNetCrossRefMATHGoogle Scholar - 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.Unanyan, R.G., Fleischhauer, M.: A geometric phase gate without dynamical phases. Phys. Rev. A
**69**(5), 050302(R) (2004)CrossRefGoogle Scholar - 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)MathSciNetCrossRefMATHGoogle Scholar - 45.Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics. Birkhäuser, Basel (1998)MATHGoogle Scholar
- 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.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.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)CrossRefMATHGoogle Scholar