Exact solutions of a quantum system placed in a Kratzer potential and under a uniform magnetic field
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Exact solutions of a quantum system placed in a Kratzer potential and under a uniform magnetic field F MAIZ1,2
,∗
and MOTEB M ALQAHTANI1
1 Faculty
of Science, Physics Department, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia of Thermal Processes, Center for Energy Research and Technology, Borj-Cedria, BP: 95, Tunisia ∗ Corresponding author. E-mail: [email protected] 2 Laboratory
MS received 8 June 2020; revised 12 August 2020; accepted 8 September 2020 Abstract. We propose Whittaker function approach as a theoretical method for finding exact solutions of a quantum mechanical system placed in the Kratzer potential. We then show that the effect of an external uniform magnetic field on this system can be satisfactorily determined using variational method. By following the one-step treatment suggested in this study, we increase the reliability and the accuracy of the solutions of Schrödinger equation for a quantum mechanical system placed in potential energy and perturbed by a uniform magnetic field that proves to be useful in modelling physical phenomena. We find that the achieved numerical and analytical results agree very well with those already published and those calculated using the Numerov method. Keywords. Kratzer potential; variational method; Numerov method; energy eigenvalues; Landau problem; Whittaker functions. PACS Nos 03.65.Ge; 31.15.xt; 03.65.Fd
1. Introduction The fundamental equation of quantum mechanics is the Schrödinger equation [1]. The solution for this equation is the wave function, which contains all the information that can be known about a quantum system. Some of the simplest systems that have been exactly solved are: the hydrogen atom, infinite square well, harmonic oscillator and quantum box. Solutions become difficult when the system consists of two or more particles and the number of exact solutions is very limited. To solve this problem, there are some approximate approaches that can analytically provide accepted solutions such as the Wentzel–Kramers–Brillouin (WKB), the variational method (VM) and the perturbation theory. Moreover, many numerical methods, such as the Airy function approach, the asymptotic iteration, the Numerov method (NM) and the finite element method [2–11] have been suggested as solutions. Finding exact solutions to the Schrödinger equation for potentials that prove useful in the modelling of physical phenomena is a very important challenge for a deep understanding of the structures and interactions in such systems. In quantum chemistry and molecular physics, Kratzer’s potential is often considered to explain interactions in a molecular system. 0123456789().: V,-vol
This potential is exactly solvable [12], and numerically solved by many methods, such as the asymptotic iteration method (AIM) [13–22]. This potential is widely used. For example, it was presented in the study of anharmonic oscillatory systems having potential energy of Kratzer type [23,24]. Relativistic and non-relativistic treatment of Hulthen–Kratzer pot
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