EXACT WAVE FUNCTIONS AND DYNAMICAL SYMMETRY GROUP FOR THE MODIFIED NONRELATIVISTIC RING-SHAPED KRATZER POTENTIAL
Shakir M. Nagiyev, Könül Sh. Jafarova, Shovqiyya A. Amirova, Vefa A. Tarverdiyeva,
2024   03   en   p.03-08

ABSTRACT

We propose a new exactly solvable potential which consists of the modified Kratzer potential plus a new ring-shaped potential ((β+γcosθ+νcos2 θ))/(r2sin2θ). The exact solutions of the bound states of the Schrödinger equation for this potential are presented analytically by using the functional method. The wavefunctions of the radial and angular parts are taken on the form of the Laguerre polynomials and the Jacobi polynomials, respectivly. Total energy of the system is different from the modified Kratzer potential because of the contribution of the angular part. We also build a dynamical symmetry group for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

Keywords: Schrödinger equation; ring-shaped modified Kratzer potential; dynamical symmetry group; Laguerre and Jacobi polynomials.
PACS: 03.65.Ge

DOI:-

Received: 08.07.2024
Internet publishing: 14.08.2024

AUTHORS & AFFILIATIONS

Institute of Physics Ministry of Science and Education Republic of Azerbaijan, 131 H.Javid ave, Baku, AZ-1143, Azerbaijan
E-mail: shakir.m.nagiyev@gmail.com, konule2016@gmail.com, shovqiyya@mail.ru, vefa.tarverdiyeva@mail.ru

 REFERENCIES

[1]   L.D. Landau, E.M. Lifshitz. Quantum Mechanics 3rd edn. (Pergamon Press, 1979).
[2]   A.S. Davydov. Quantum Mechanics (Pergamon Press, 1965).
[3]   W. Greiner. Relativistic Quantum Mechanics 3rd edn. (Springer, 2000).
[4]   S. Flügge. Practical Quantum Mechanics Vol. 1 (Springer, 1994).
[5]   V.G. Bagrov, D.M. Gitman. Exact Solutions of Relativistic Wave Equations (Kluwer Academic Publishers, 1990).
[6]   A. Hautot. Exact motion in noncentral electric felds. J. Math. Phys. 14(10), 1320–1327 (1973).
[7]   Sh.M. Nagiyev, A.I. Ahmadov. Exact solution of the relativistic fnite diference equation for the Coulomb plus Ring-Shaped potential. Int. J. Mod. Phys. A 34(17), 1950089 (2019).
[8]   A. Kratzer. Die ultraroten Rotationsspektren der Halogenwasserstofe. Z. Phys. 3(5), 289–307 (1920).
[9]   H. Hartmann. Die Bewegung eines Körpers in einem ringförmigen Potentialfeld. Teor. Chim. Acta 24(2–3), 201–206 (1972).
[10]  C.J. Quesne. A new ring-shaped potential and its dynamical invariance algebra. Phys. A: Math. Gen. 21, 3093–3103 (1988).
[11]  De Souza Dutra, G. Chen. On some classes of exactly-solvable Klein–Gordon equations, Phys. Lett. A 349 (2006) 297 – 301.
[12]  C. Berkdemir, A. Berkdemir and J.G. Han. Bound state solutions of the Schrödinger equation for modified Kratzer molecular potential, Chem. Phys. Lett. 417, 2006, 326.
[13]  Y.F. Cheng, T.Q. Dai. Exact solution of the Schrödinger equation for modified Kratzer potential plus a ring-shaped potential by the Nikiforov – Uvarov methad, Phys. Scr. 75 (2007) 274 – 277.
[14]  C.Y. Chen, S.H. Dong. Exactly complete solutions of the Couloumb potential plus a new ring-shaped potential, Phys. Lett. A 335 (2005) 374-382.
[15]  C.Y. Fu, D.T. Qing. Exact solutions of the Schrödinger equation for a new ring-shaped nonharmonic oscillator potential. Int. J. Mod. Phys. A 23 (2008) 1919-1927.
[16]  K.J. Oyewumi. Analytical solutions of the Kratzer-Fues potential in an arbitrary number of dimensions, Found. Phys. Lett. 18 (2005) 75 – 84.
[17]  A. Durmus, F. Yasuk. Relativistic and nonrelativistic solutions for diatomic molecules in the ppresence of double ring-shaped Kratzer potential, J. Chem. Phys. 126, 074108 (2007).
[18]  Sh.M. Nagiyev, A.I. Ahmadov, A.N. Ikot, V.A.Tarverdiyeva. Analytic solutions for the Klein-Gordon equation with combined exponential type and ring-shaped potentials, Scientific Rep., (2024) 145527.
[19]  S.M. Nagiyev, A.I. Ahmadov, V.A. Tarverdiyeva. Approximate solutions to the Klein–Fock–Gordon equation for the sum of coulomb and ring-shaped-like potentials. Adv. High Energy Phys. 2020, 1356384 (2020).
[20]  R. Koekoek, P.A. Lesky and R.F. Swarttow. Hypergeometric Orthogonal Polynomials and their q-Analogues (Springer-Verlag, Berlin, 2010).
[21]  A.O. Barut and R. Raczka. Theory of Group Representations and Applications (Polish Scientific Publishers, Warszawa, 1977.