† Corresponding author. E-mail:
We need to solve a suitable exponential form of the position-dependent mass (PDM) Schrödinger equation with a charged particle placed in the Hulthen plus Coulomb-like potential field and under the actions of the external magnetic and Aharonov–Bohm (AB) flux fields. The bound state energies and their corresponding wave functions are calculated for the spatially-dependent mass distribution function of interest in physics. A few plots of some numerical results with respect to the energy are shown.
The exact solutions of quantum wave equations, expressed in analytical form, describing one-electron atoms and few-body systems are essential in studying the atomic structure theory and more areas. In fact, the exact analytical solutions are essentially used in quantum-chemical, quantum electrodynamics and theory of molecular vibrations. They are also used to examine the correctness of models, approximations in computational physics, nuclear physics, nanostructures and computational chemistry.[1–16]
On the other hand, the position-dependent mass (PDM) idea arises after the effect of the periodic field on the non-relativistic motion of electrons in periodic lattices. In fact, it happens in typical semiconductors by the effects of impurities in perturbed periodic lattices.[17] Recently, a considerable interest in the mass dependence on the internuclear distance has been revived in solving different equations with various physical potential models.[18–29] Furthermore, a number of studies take the effect of an electric field or a magnetic field into account in studying different systems.[30–34] For further study we cite Refs. [35] and [36]. In addition, it is found to be more interesting that nearly all desired analytic solutions of the non-relativistic equation have been expressed in terms of hypergeometric functions.[38–40]
However, in all these areas, the studies of the non-relativistic and relativistic quantum dynamics of charged particles in the presence of magnetic fields and Aharonov–Bohm (AB) flux fields, which are perpendicular to the plane where the particles are confined, have been carried out over the past few years.[41] In fact, the investigation of systems consisting of non-relativistic as well as relativistic charged particles, that are confined by the magnetic fields, has attracted a great deal of attention due to their applications (such as in graphene,[15,16,42] semiconductor structures,[43] chemical physics,[44] molecular vibrational and rotational spectroscopy of molecular physics,[45] biology,[46] environmental sciences,[47] and cosmic string[48]). Recently, Eshghi et al.[49] solved the Schrödinger equation with the superposition of Morse-plus-Coulomb potentials with two different physically PDM distribution functions of the exponential and inverse-square forms in the external perpendicular magnetic and AB flux fields. Also, these authors have investigated the Schrödinger equation with a position-dependent mass charged particle interacted via the superposition of the Morse-plus-Coulomb potentials under the actions of external magnetic and Aharonov–Bohm flux fields.[49] Also, Jiang et al. have investigated the solutions of the Schrödinger equation with a position-dependent mass.[50] However, in the present work, we intend to extend the work in Ref. [51] and solve the Schrödinger equation with a charged particle under the actions of Hulthen plus Coulomb-like potential field having the general form:
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Figures
For example, figure
For
The rest of this paper is organized as follows. In section
In this section, we are to solve the Schrödinger equation for a charged particle with a physically position-dependent mass (PDM) distribution function interacted via the Hulthen plus Coulomb-like potential field and exposed to external perpendicular magnetic and AB flux fields treated in two-dimensional space cylindrical coordinates. We are to calculate the bound state energies and their corresponding wave functions. The general form of the Schrödinger equation for a charged particle with the PDM system under the action of a certain potential field and in the presence of the vector potential is given by
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![](cpb_27_2_020301/cpb_27_2_020301_ieqn11.gif)
![](cpb_27_2_020301/cpb_27_2_020301_ieqn12.gif)
We assume that the vector potential has the simple form:
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![](cpb_27_2_020301/cpb_27_2_020301_ieqn13.gif)
Substituting the vector potential (
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Now using the parameterized Nikoforov–Uvarov method[53] with the following substitution
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Therefore, it is obvious from Fig.
In Figs.
To proceed in our discussion, we can use energy spectrum formula (
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![](cpb_27_2_020301/cpb_27_2_020301_ieqn19.gif)
![](cpb_27_2_020301/cpb_27_2_020301_ieqn20.gif)
On the other hand, after using Eq. (
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We solved the Schrödinger equation for a charged particle with an exponential form position-dependent mass (PDM) placed in the field of the general superposition of Húlthen plus Coulomb-like potential fields under the actions of external magnetic and AB flux fields. We calculated the bound state energies and the corresponding wave functions with a suitable change with respect to the spatially dependent mass variable as functions of the magnetic and AB flux fields by using the parameterized NU method. Finally, some results of the energy values are shown in Figs.
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