Thermomagnetic properties and its effects on Fisher entropy with Schioberg plus Manning-Rosen potential (SPMRP) using Nikiforov-Uvarov functional analysis (NUFA) and supersymmetric quantum mechanics (SUSYQM) methods

Abstract

Thermomagnetic properties, and its effects on Fisher information entropy with Schioberg plus Manning-Rosen potential are studied using NUFA and SUSYQM methods in the presence of the Greene-Aldrich approximation scheme to the centrifugal term. The wave function obtained was used to study Fisher information both in position and momentum spaces for different quantum states by the gamma function and digamma polynomials. The energy equation obtained in a closed form was used to deduce numerical energy spectra, partition function, and other thermomagnetic properties. The results show that with an application of AB and magnetic fields, the numerical energy eigenvalues for different magnetic quantum spins decrease as the quantum state increases and completely removes the degeneracy of the energy spectra. Also, the numerical computation of Fisher information satisfies Fisher information inequality products, indicating that the particles are more localized in the presence of external fields than in their absence, and the trend shows complete localization of quantum mechanical particles in all quantum states. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. The energy equations obtained from the NUFA and SUSYQM were the same, demonstrating a high level of mathematical precision.

Introduction

Over the years, devices that enable quantum information to be coherently transferred between topological and conventional materials have been studied by various researchers¹. These materials contained an electromagnetic field, which serves as a fundamental carrier of information, capable of transmitting a modulated signal and collecting data about the propagation channel itself². This was made possible by the foundations of information theory by Fisher³ in his classical measurement theory, which he used for estimating ultimate quantum limits that allow for known local changes in density⁴. The context was also examined by Shannon⁵ in his study. The Shannon entropy is a global measure of electron density that plays a significant role in the assessment of uncertainty and provides a source of information about atomic, molecular, and nuclear systems^6,7,8.

These theoretic tools provide a deeper understanding of density functional and electron correlation in studying the structure and dynamics of the atomic system⁹. Quantum information theory has proven to be extremely useful in a wide range of fields, such as Physics, Chemistry, Biology, Medicine, Computer science, neural networks, linguistics, and other social sciences¹⁰. They are commonly used in quantum physics to analyze quantum steering¹¹, quantum entanglement¹², quantum revivals¹³, quantum communication¹⁴, atomic ionization properties¹⁵, and other phenomena. In wave mechanics, the solutions of the eigenfunctions of the Schrödinger equation under a potential energy barrier are essential because the entropic functionals are presented in terms of probability densities in the position and momentum spaces¹⁶. Several research have been carried out on Shannon entropy and Fisher information with physically motivated potential models, like the class of Yukawa potential¹⁷, Screened Coulomb potential⁹, generalized hyperbolic potential¹⁸, screened Kratzer potential¹⁹, Frost-Musulin potential²⁰, hyperbolic potential²¹, and many others.

The Manning-Rosen potential is a significant exponential-type potential proposed by Manning and Rosen²² in 1933 to explain the vibrational behavior of the model of the diatomic molecule²³. The form of this potential model is given by²⁴

$$V\left( r \right) = - \frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }},$$

(1)

where $c_{1} {\text{ and }}c_{2}$ are potential strength parameters, and $\alpha$ represents the screening parameter while $r$ is the inter-particle distance. The Schioberg potential, proposed in 1986, is another intriguing potential. This potential describes the molecular vibrations of diatomic molecules accurately more than the Morse function and represents suitably intermolecular interactions between particles²⁵. The potential model is of the form²⁶

$$V\left( r \right) = D\left[ {1 - \delta_{0} \left( {\frac{{1 + e^{ - \alpha r} }}{{1 - e^{ - \alpha r} }}} \right)} \right]^{2} ,$$

(2)

where $D$ is the potential depth, $\alpha$ is the screening parameter and $\delta_{0}$ is the potential parameter that determines the size of the potential and can also serve as optimizing parameter. Recently, there has been a surge of interest in incorporating at least two potentials. The goal of combining at least two potential models is to provide more physical application and analysis to existing molecular physics studies. Also, it is well-known that the potential energy functions with more parameters have a tendency to fit experimental data better than those with fewer parameters^27,28,29. Many scholars have conducted extensive research in both relativistic and non-relativistic regimes to explore these potentials^{30,31,32,33,34,35,36,37,38}.

In recent times, research indicates that the addition of external fields to potential functions on quantum systems has demonstrated its potency in controlling certain behaviors of systems and molecules³⁹. The Aharonov–Bohm (AB) effect, discovered in 1961⁴⁰, occurs when a moving charge is transformed by scalar and vector potentials that appear in the Schrödinger equation (SE) even in the absence of external EM fields⁴¹. Since then, many studies have analyzed a bound state of a charged particle moving in a potential vector and scalar potential. A realistic description of the external EM field effects on quantum systems is provided by the Stark⁴² and Zeeman⁴³ effects. In the Stark effects, an external electric field is applied to the electrically neutral hydrogen atom, causing it to experience a zero net force, resulting in a shift in the energy levels. On the other hand, Zeeman effects occur when an atom is exposed to a uniform magnetic field. These interactions have similar effects in that they cause the energy levels to split and shift⁴⁴. External fields have previously been studied by a wide range of quantum mechanical phenomena in many areas, including physics, chemistry, biology, material science, engineering, mathematics^{45,46,47,48,49,50,51} and others.

Considering the vast applicability of the Manning-Rosen and Schioberg potentials, it is necessary to investigate the bound state solutions of the two-dimensional (2D) SE with the combined potential under the influence of external magnetic and Aharonov–Bohm (AB) fields and their effects on the Shannon entropy and Fisher information for some selected diatomic molecules. The bound state solutions will be obtain using the Nikiforov-Uvarov-Functional Analysis (NUFA) and supersymmetric quantum mechanics (SUYSQM) methods.

This paper is organized as follows: first, we provide detailed solutions to the 2D SE with Manning-Rosen plus Schioberg potential (SPMR) in the presence of magnetic and Aharonov–Bohm (AB) flux fields using the NUFA method. Second, we used the SUYSQM method to obtain the analytical solution of the SE with the combined potential in the presence of magnetic and Aharonov–Bohm (AB) flux fields. Also, the normalized wavefunction obtained is applied to investigate the Shannon entropy and Fisher information in the presence and absence of external magnetic and Aharonov–Bohm (AB) flux fields. Finally, the concluding remarks. The SPMR is of the form.

$$V\left( r \right) = D\left[ {1 - \sigma_{0} \left( {\frac{{1 + e^{ - \alpha r} }}{{1 - e^{ - \alpha r} }}} \right)} \right]^{2} - \left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right)$$

(3)

Nikiforov-Uvarov-Functional Analysis (NUFA) method

The Nikiforov-Uvarov Functional Analysis (NUFA) method recently developed by Ikot et al.⁵² has been very helpful in providing solutions for exponential type potentials both in relativistic and nonrelativistic wave equations When using this method to solve either the Schrödinger or Klein–Gordon equation, the energy eigen equation is directly presented in a factorized, closed and compact form. This gives the method an edge over other methods. Meanwhile, the NUFA theory involves solving second order Schrödinger-like differential equation through the analytical combination of Nikiforov-Uvarov (NU) method and functional analysis approach^53,54,55. NU is applied to solve a second-order differential equation of the form

$$\frac{{d^{2} {\uppsi }\left( {\text{s}} \right)}}{{ds^{2} }} + \frac{{\tilde{\tau }\left( s \right)}}{\sigma \left( s \right)}\frac{{d{\uppsi }\left( {\text{s}} \right)}}{ds} + \frac{{\tilde{\sigma }\left( s \right)}}{{\sigma^{2} \left( s \right)}}\psi \left( s \right) = 0$$

(4)

where $\sigma (s)$ and $\widetilde{\sigma }\left(s\right)$ are polynomials at most degree two and $\widetilde{\tau }(s)$ is a first-degree polynomial. Tezean and Sever⁵⁶ latter introduced the parametric form of NU method in the form

$$\frac{{d^{2} \psi (s)}}{{ds^{2} }} + \frac{{\alpha_{1} - \alpha_{2} s}}{{s(1 - \alpha_{3} s)}}\frac{{d^{2} \psi (s)}}{{ds^{2} }} + \frac{1}{{s^{2} (1 - \alpha_{3} s)^{2} }}\left[ { - U_{1} s^{2} + U_{2} s - U_{3} } \right]\psi (s) = 0,$$

(5)

where $\alpha_{i}$ and $\xi_{i} (i = 1,2,3)$ are all parameters. The differential Eq. (3) has two singularities which is at $s \to 0$ and $s \to \frac{1}{{\alpha_{3} }}$ thus, the wave function can be expressed in the form.

$$\Psi_{n} (s) = s^{\lambda } (1 - \alpha_{3} s)^{v} f(s)$$

(6)

Substituting Eq. (6) into Eq. (5) and simplifying culminate to the following equation,

$$\begin{aligned} s(1 - \alpha_{3} s)\frac{{d^{2} f(s)}}{{ds^{2} }} & + \left[ {\alpha_{1} + 2\lambda - (2\lambda \alpha_{3} + 2v\alpha_{3} + \alpha_{2} )s} \right]\frac{df(s)}{{ds}} \\ & - \alpha_{3} \left( {\lambda + v + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}} } \right)\\&\quad \left( {\lambda + v + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) - \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}} } \right)f\left( s \right) \\ & + \left[ {\frac{{\lambda (\lambda - 1) + \alpha_{1} \lambda - U_{3} }}{s} + \frac{{v(v - 1)\alpha_{3} + \alpha_{2} v - \alpha_{1} \alpha_{3} v - \frac{{U_{1} }}{{\alpha_{3} }} + U_{2} - U_{3} \alpha_{3} }}{{\left( {1 - \alpha_{3} s} \right)}}} \right]f\left( s \right) = 0 \\ \end{aligned}$$

(7)

Equation (7) can be reduced to a Guassian- hypergeometric equation if and only if the following functions vanished

$$\lambda \left( {\lambda - 1} \right) + \alpha_{1} \lambda - U_{3} = 0$$

(8)

$$\upsilon \left( {\upsilon - 1} \right)\alpha_{3} + \alpha_{2} \upsilon - \alpha_{1} \alpha_{3} \upsilon - \frac{{U_{1} }}{{\alpha_{3} }} + U_{2} - U_{3} \alpha_{3} = 0.$$

(9)

Applying the condition of Eq. (8) and Eq. (9) into Eq. (7) results into Eq. (10)

$$\begin{aligned} s(1 - \alpha_{3} s) & \frac{{d^{2} f(s)}}{{ds^{2} }}\left[ {\alpha_{1} + 2\lambda - (2\lambda \alpha_{3} + 2v\alpha_{3} + \alpha_{2} )s} \right]\frac{df(s)}{{ds}} \\ & \;\; - \alpha_{3} \left( {\lambda + v + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}} } \right)\\ & \quad\left( {\lambda + v + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) - \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}} } \right)f\left( s \right) = 0 \\ \end{aligned}$$

(10)

The solutions of Eqs. (8) and (9) are given as

$$\lambda = \frac{1}{2}\left( {\left( {1 - \alpha_{1} } \right) \pm \sqrt {\left( {1 - \alpha_{1} } \right)^{2} + 4U_{3} } } \right)$$

(11)

$$\upsilon = \frac{1}{{2\alpha_{3} }}\left( {\left( {\alpha_{3} + \alpha_{1} \alpha_{3} - \alpha_{2} } \right) \pm \sqrt {\left( {\alpha_{3} + \alpha_{1} \alpha_{3} - \alpha_{2} } \right)^{2} + 4\left( {\frac{{U_{1} }}{{\alpha_{3} }} + \alpha_{3} U_{3} - U_{2} } \right)} } \right)$$

(12)

Equation (10) is the hypergeometric equation type of the form

$$x\left( {1 - x} \right)\frac{{d^{2} f(s)}}{{ds^{2} }} + \left[ {c - \left( {a + b + 1} \right)x} \right]\frac{df(x)}{{dx}} - \left[ {ab} \right]f(x) = 0$$

(13)

where a, b and c are given as follows:

$$a = \sqrt {\alpha_{3} } \left( {\lambda + v + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{U_{3}^{2} }}} } \right)$$

(14)

$$b = \sqrt {\alpha_{3} } \left( {\lambda + v + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) - \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}} } \right)$$

(15)

$$c = \alpha_{1} + 2\lambda$$

(16)

Setting either a or b equal to a negative integer – n, the hypergeometric function f(s) turns to a polynomial of degree n. Hence, the hypergeometric function f(s) approaches finite in the following quantum condition, i.e.,$a = - n$ where $n = 0,1,2,3 \ldots n_{\max }$ or $b = - n$.

Using the above quantum condition,

$$\sqrt {\alpha_{3} } \left( {\lambda + \upsilon + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}} } \right) = - n$$

(17)

$$\lambda + \upsilon + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \frac{n}{{\sqrt {\alpha_{3} } }} = - \sqrt {\frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} + \frac{{U_{1} }}{{\alpha_{3}^{2} }}}$$

(18)

By simplifying Eq. (18), the energy eigen equation using NUFA method is given as

$$\lambda^{2} + 2\lambda \left( {\upsilon + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \frac{n}{{\sqrt {\alpha_{3} } }}} \right) + \left( {\upsilon + \frac{1}{2}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right) + \frac{n}{{\sqrt {\alpha_{3} } }}} \right)^{2} - \frac{1}{4}\left( {\frac{{\alpha_{2} }}{{\alpha_{3} }} - 1} \right)^{2} - \frac{{U_{1} }}{{\alpha_{3}^{2} }} = 0$$

(19)

By substituting Eqs. (9) and (10) into Eq. (6), the corresponding wave equation for the NUFA method as

$$\Psi_{n} (s) = N_{n} S^{{\frac{{\left( {1 - \alpha_{1} } \right) + \sqrt {\left( {\alpha_{1} - 1} \right)^{2} + 4U_{3} } }}{2}}} \left( {1 - \alpha_{3} } \right)^{{\frac{{\left( {\alpha_{3} + \alpha_{1} \alpha_{3} - \alpha_{2} } \right) + \sqrt {\left( {\alpha_{3} + \alpha_{1} \alpha_{3} - \alpha_{2} } \right)^{2} + 4\left( {\frac{{U_{1} }}{{\alpha_{3}^{2} }} + \alpha_{3} U_{3} - U_{2} } \right)} }}{{2\alpha_{2} }}}} {}_{2}F_{1} (a,b,c;s)$$

(20)

Thermomagnetic energy spectra of 2-dimensional Schrödinger equation under the influence Aharanov-Bohm (AB) flux and external magnetic field using NUFA

The thermomagnetic energy spectra of 2-Dimensional Schrödinger equation under the influenced of AB and Magnetic field with SPMR potential can be obtained from charged particle Hamiltonian operator of the form

$$\left\{ {\frac{1}{2\mu }\left( {i\hbar \nabla - \frac{e}{c}\vec{A}} \right)^{2} + D\left[ {1 - \sigma_{0} \left( {\frac{{1 + e^{ - \alpha r} }}{{1 - e^{ - \alpha r} }}} \right)} \right]^{2} - \left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right)} \right\}R\left( {r, \varphi } \right) = E_{nm} R\left( {r, \varphi } \right)$$

(21)

$E_{nm}$ is the thermomagnetic energy spectra, $e$ and $\mu$ represent the charge of the particle and the reduced mass respectively. $c$ is the speed of light. Meanwhile, The vector potential $\overrightarrow{A}=\left({A}_{r},{A}_{\phi }, {A}_{z}\right)$ can be written as the superposition of two terms such that $\overrightarrow{A}=\overrightarrow{{A}_{1}}+\overrightarrow{{A}_{2}}$ is the vector potential with azimuthal components such that $\overrightarrow{{A}_{1}}=$ and $\overrightarrow{{A}_{2}}=$, corresponding to the extra magnetic flux $\Phi_{AB}$ generated by a solenoid with $\overrightarrow{\nabla }.\overrightarrow{{A}_{2}}=0$ and $\overrightarrow{B}$ is the magnetic vector field accompanied by $\overrightarrow{\nabla }\times \overrightarrow{{A}_{1}}=\overrightarrow{B}$ ,$\overrightarrow{\nabla }\times \overrightarrow{{A}_{2}}=0$. The vector potential $\overrightarrow{A}$ can then be expressed as

$$\vec{A} = \left( {0,\frac{{Be^{ - \alpha r} \hat{\varphi }}}{{1 - e^{ - \alpha r} }} + \frac{{\Phi_{AB} }}{2\pi r}\hat{\varphi },0} \right) = \left( {\frac{{Be^{ - \alpha r} \hat{\varphi }}}{{1 - e^{ - \alpha r} }} + \frac{{\Phi_{AB} }}{2\pi r}\hat{\varphi }} \right)$$

(22)

The Laplacian operator and the wave function in cylindrical coordinate is given as

$$\begin{aligned} \nabla^{2} & = \frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }} \\ \Psi \left( {r,\varphi } \right) & = \frac{1}{{\sqrt {2\pi r} }}R_{nm} (r)e^{im\varphi } \\ \end{aligned}$$

(23)

where $m$ represents the magnetic quantum number. Substituting Eqs. (23) and (22) into Eq. (21) and with much algebraic simplification gives rise to the Schrödinger -like equation of the form

$$\frac{{d^{2} R_{nm} (r)}}{{dr^{2} }} + \frac{2\mu }{{h^{2} }}\left[ \begin{gathered} E_{nm} - D\left[ {1 - \sigma_{0} \left( {\frac{{1 + e^{ - \alpha r} }}{{1 - e^{ - \alpha r} }}} \right)} \right]^{2} + \left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) - \hbar \omega_{c} \left( {m + \xi } \right)\frac{{e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)r}} \hfill \\ - \left( {\frac{{\mu \omega_{c}^{2} }}{2}} \right)\frac{{e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} - \frac{{\hbar^{2} }}{2\mu }\left( {\frac{{\left( {m + \xi } \right)^{2} - \frac{1}{4}}}{{r^{2} }}} \right) \hfill \\ \end{gathered} \right]R_{nm} (r) = 0.$$

(24)

where $\xi = \frac{{\Phi_{AB} }}{{\phi_{0} }}$ is an absolute value containing the flux quantum $\phi_{0} = \frac{hc}{e}$. The cyclotron frequency is represented by $\omega_{c} = \frac{{e\vec{B}}}{\mu c}$. Equation (24) is not exactly solvable due to the presence of centrifugal barrier $\frac{1}{{r^{2} }}$. In order to provide an analytical approximate solution to Eq. (24), we substitute the modified Greene-Aldrich approximation of the form $\frac{1}{{r^{2} }} = \frac{{\alpha^{2} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}$ into Eq. (24) to deal with the centrifugal barrier. Also, using the coordinate transformation $s = e^{ - \alpha r}$ together with the approximation term, Eq. (24) reduced to the hyper-geometric equation of the form

$$\frac{{d^{2} R_{nm} (s)}}{{ds^{2} }} + \frac{{\left( {1 - s} \right)}}{{s\left( {1 - s} \right)}}\frac{{dR_{nm} (s)}}{ds} + \frac{1}{{s^{2} \left( {1 - s} \right)^{2} }}\left\{ \begin{gathered} - \left( {\varepsilon^{2} + \chi_{1} + 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} - \chi_{2} + \chi_{5} } \right)s^{2} \hfill \\ + \left( {2\varepsilon^{2} + 2\chi_{1} - 2\chi_{1} \sigma_{0}^{2} + \chi_{3} - \chi_{4} } \right)s \hfill \\ - \left( {\varepsilon^{2} + \chi_{1} - 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} + \chi_{6} } \right) \hfill \\ \end{gathered} \right\}R_{nm} (s) = 0.$$

(25)

where

$$\begin{aligned} \varepsilon^{2} & = - \frac{{2\mu E_{nm} }}{{\hbar^{2} \alpha^{2} }}\begin{array}{*{20}c} , & {\chi_{1} = \frac{2\mu D}{{\hbar^{2} \alpha^{2} }}} \\ \end{array} \begin{array}{*{20}c} , & {\chi_{2} = \frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }}} \\ \end{array} \begin{array}{*{20}c} , & {\chi_{3} = \frac{{2\mu c_{2} }}{{\hbar^{2} \alpha^{2} }}} \\ \end{array} \\ \chi_{4} & = \frac{{2\mu \omega_{c} \left( {m + \xi } \right)}}{\hbar \alpha }\begin{array}{*{20}c} , & {\chi_{5} = \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }}} \\ \end{array} \begin{array}{*{20}c} , & {\chi_{6} = \left( {m + \xi } \right)^{2} - \frac{1}{4}} \\ \end{array} . \\ \end{aligned}$$

(26)

Comparing Eq. (25) with NUFA differential equation in Eq. (5), the following polynomial equations can be obtained.

$$\begin{gathered} U_{1} = \left( {\varepsilon^{2} + \chi_{1} + 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} - \chi_{2} + \chi_{5} } \right)\begin{array}{*{20}c} , & {U_{2} = \left( {2\varepsilon^{2} + 2\chi_{1} - 2\chi_{1} \sigma_{0}^{2} + \chi_{3} - \chi_{4} } \right)} \\ \end{array} \hfill \\ U_{3} = \left( {\varepsilon^{2} + \chi_{1} - 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} + \chi_{6} } \right),\alpha_{1} = \alpha_{2} = \alpha_{3} = 1. \hfill \\ \end{gathered}$$

(27)

Using equation NUFA in Eqs. (11), (12), (14), (15) and (16) the following polynomial equations can be obtained

$$\lambda = \sqrt {\varepsilon^{2} + \chi_{1} - 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} + \chi_{6} } ,$$

(28)

$$\upsilon = \frac{1}{2} + \frac{1}{2}\sqrt {16\chi_{1} \sigma_{0}^{2} - 4\chi_{2} + 4\chi_{5} + 4\chi_{6} - 4\chi_{3} + 4\chi_{4} + 1} ,$$

(29)

$$a = \left( \begin{gathered} \sqrt {\varepsilon^{2} + \chi_{1} - 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} + \chi_{6} } + \frac{1}{2} + \frac{1}{2}\sqrt {16\chi_{1} \sigma_{0}^{2} - 4\chi_{2} + 4\chi_{5} + 4\chi_{6} - 4\chi_{3} + 4\chi_{4} + 1} \hfill \\ + \sqrt {\varepsilon^{2} + \chi_{1} + 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} - \chi_{2} + \chi_{5} } \hfill \\ \end{gathered} \right),$$

(30)

$$b = \left( \begin{gathered} \sqrt {\varepsilon^{2} + \chi_{1} - 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} + \chi_{6} } + \frac{1}{2} + \frac{1}{2}\sqrt {16\chi_{1} \sigma_{0}^{2} - 4\chi_{2} + 4\chi_{5} + 4\chi_{6} - 4\chi_{3} + 4\chi_{4} + 1} \hfill \\ - \sqrt {\varepsilon^{2} + \chi_{1} + 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} - \chi_{2} + \chi_{5} } \hfill \\ \end{gathered} \right),$$

(31)

$$c = \left( {1 + 2\sqrt {\varepsilon^{2} + \chi_{1} - 2\chi_{1} \sigma_{0} + \chi_{1} \sigma_{0}^{2} + \chi_{6} } } \right).$$

(32)

using Eq. (19), the thermo-magnetic energy eigen equation

$$\begin{aligned} \varepsilon^{2} & = \frac{1}{4}\left\{ {\frac{{\left( {n + \frac{1}{2} + \frac{1}{2}\sqrt {16\chi_{1} \sigma_{0}^{2} - 4\chi_{2} + 4\chi_{5} + 4\chi_{6} - 4\chi_{3} + 4\chi_{4} + 1} } \right)^{2} + \chi_{2} - \chi_{5} + \chi_{6} - 4\chi_{1} \sigma_{0} }}{{\left( {n + \frac{1}{2} + \frac{1}{2}\sqrt {16\chi_{1} \sigma_{0}^{2} - 4\chi_{2} + 4\chi_{5} + 4\chi_{6} - 4\chi_{3} + 4\chi_{4} + 1} } \right)}}} \right\}^{2} \\ & \;\;\; + 2\chi_{1} \sigma_{0} - \chi_{1} - \chi_{1} \sigma_{0}^{2} - \chi_{6} \\ \end{aligned}$$

(33)

Substituting the parameters of Eq. (26) into Eq. (33), the thermomagnetic energy equation become

$$\begin{aligned} E_{nm} & = \frac{{h^{2} \alpha^{2} }}{2\mu }\left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) + D\left( {\sigma_{0} - 1} \right)^{2} \\ & \;\;\; - \frac{{h^{2} \alpha^{2} }}{8\mu }\left\{ {\frac{\begin{gathered} \left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]^{2} \hfill \\ + \frac{{2\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4} \hfill \\ \end{gathered} }{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]}}} \right\}^{2} \\ \end{aligned}$$

(34)

The 2D nonrelativistic energy eigen equation can be obtained with the condition that $\omega_{c} = \xi = 0$, $m = l + \frac{1}{2}$.

Then Eq. (34) become

$$\begin{aligned} E_{nm} & = \frac{{h^{2} \alpha^{2} l\left( {l + 1} \right)}}{2\mu } + D\left( {\sigma_{0} - 1} \right)^{2} \\ & \;\;\; - \frac{{h^{2} \alpha^{2} }}{8\mu }\left\{ {\frac{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt {1 + \frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + 4l\left( {l + 1} \right)} } \right]^{2} + \frac{{2\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{h^{2} \alpha^{2} }} + l\left( {l + 1} \right)}}{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt {1 + \frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + 4l\left( {l + 1} \right)} } \right]}}} \right\}^{2} \\ \end{aligned}$$

(34a)

Special cases

Schioberg potential

Substituting $c_{1} = c_{2} = 0$.into Eq. (3), then, the potential reduces to Schioberg potential given as

$$V\left( r \right) = D\left[ {1 - \sigma_{0} \left( {\frac{{1 + e^{ - \alpha r} }}{{1 - e^{ - \alpha r} }}} \right)} \right]^{2} .$$

(34b)

Substituting the same condition to Eq. (34) gives the energy-eigen equation for Schioberg potential under the influence of magnetic and AB field as

$$\begin{aligned} E_{nm} & = \frac{{h^{2} \alpha^{2} }}{2\mu }\left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) + D\left( {\sigma_{0} - 1} \right)^{2} \\ & \;\;\; - \frac{{h^{2} \alpha^{2} }}{8\mu }\left\{ {\frac{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]^{2} - \frac{{\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}}}{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]}}} \right\}^{2} \\ \end{aligned}$$

(34c)

Manning-Rosen potential

Substituting $D = 0$ into Eq. (3), then the potential reduces to Manning-Rosen potential of the form

$$V\left( r \right) = - \left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right)$$

(34d)

Substituting the same condition to Eq. (34) gives the energy eigen equation of Manning-Rosen potential under the influence of magnetic and AB fields as

$$\begin{aligned} E_{nm} & = \frac{{h^{2} \alpha^{2} }}{2\mu }\left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) \\ & \;\;\; - \frac{{h^{2} \alpha^{2} }}{8\mu }\left\{ {\frac{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt { - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]^{2} + \frac{{2\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}}}{{\left[ {n + \frac{1}{2} + \frac{1}{2}\sqrt { - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]}}} \right\}^{2} \\ \end{aligned}$$

(35)

Using Eq. (20), the wave function can be presented in a factorized form as

$$\Psi_{nm} (s) = N_{n} S^{\beta } \left( {1 - s} \right)^{{\eta + \frac{1}{2}}} F_{1} \left( {a,b,c;s} \right)$$

(36)

where

$$\begin{gathered} \beta = \sqrt {\frac{2\mu D}{{h^{2} \alpha^{2} }} - \frac{{2\mu E_{nm} }}{{h^{2} \alpha^{2} }} - \frac{{4\mu D\sigma_{0} }}{{h^{2} \alpha^{2} }} + \frac{{2\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \hfill \\ \eta = \sqrt {\frac{{8\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{2\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{2\mu c_{2} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi + \frac{{\mu \omega_{c} }}{h\alpha }} \right)^{2} } \hfill \\ \end{gathered}$$

(37)

Equation (36) can be expressed in terms of Jacobi polynomial as

$$\Psi_{nm} (s) = N_{n} S^{\beta } \left( {1 - s} \right)^{{\eta + \frac{1}{2}}} P_{n}^{{\left( {2\beta ,2\eta } \right)}} (1 - 2s)$$

(38)

Equation (38) can be normalized using the expression

$$\begin{gathered} \int\limits_{0}^{\infty } {\left| {\Psi_{nm} (s)} \right|^{2} } dr = 1 \hfill \\ \Rightarrow N_{nl}^{2} \int\limits_{0}^{\infty } {(e^{ - \alpha r} )^{2\beta } (1 - e^{ - \alpha r} )^{2\eta + 1} \left| {P_{n}^{{\left( {2\beta ,2\eta } \right)}} \left( {1 - 2e^{ - \alpha r} } \right)} \right|^{2} dr} = 1 \hfill \\ \end{gathered}$$

(39)

Using Mathematica 10.0 version, the normalized wave function for ground states, first excited state, second excited state and third excited quantum state can be obtained as follows:

$$\Psi_{0,m} (r) = \sqrt {\frac{{\alpha \Gamma \left( {2\beta + 2\eta + 2} \right)}}{{\Gamma \left( {2\beta } \right)\Gamma \left( {2 + 2\eta } \right)}}} \left( {e^{ - \alpha r} } \right)^{\beta } \left( {1 - e^{ - \alpha r} } \right)^{{\eta + \frac{1}{2}}}$$

(40)

$$\Psi_{1,m} (r) = \sqrt {\frac{{2\alpha \beta \Gamma \left( {3 + 2\beta + 2\eta } \right)\Gamma \left( {2\beta + 2\eta + 2} \right)}}{{\Gamma \left( {3 + 2\eta } \right)\Gamma \left( {2\beta } \right)\Gamma \left( {2 + 2\eta } \right)}}} \left( {e^{ - \alpha r} } \right)^{\beta } \left( {1 - e^{ - \alpha r} } \right)^{{\eta + \frac{1}{2}}} P_{1}^{{\left( {2\beta ,2\eta } \right)}} \left( {1 - 2e^{ - \alpha r} } \right)$$

(41)

$$\Psi_{2,m} (r) = \sqrt {\frac{{4\alpha \beta \Gamma \left( {5 + 2\beta + 2\eta } \right)\Gamma \left( {2\beta + 2\eta + 3} \right)}}{{\Gamma \left( {5 + 2\eta } \right)\Gamma \left( {3 + 2\beta } \right)\Gamma \left( {3 + 2\eta } \right)}}} \left( {e^{ - \alpha r} } \right)^{\beta } \left( {1 - e^{ - \alpha r} } \right)^{{\eta + \frac{1}{2}}} P_{2}^{{\left( {2\beta ,2\eta } \right)}} \left( {1 - 2e^{ - \alpha r} } \right)$$

(42)

$$\Psi_{3,m} (r) = \sqrt {\frac{{12\alpha \beta \Gamma \left( {7 + 2\beta + 2\eta } \right)\Gamma \left( {2\beta + 2\eta + 4} \right)}}{{\Gamma \left( {7 + 2\eta } \right)\Gamma \left( {4 + 2\beta } \right)\Gamma \left( {4 + 2\eta } \right)}}} \left( {e^{ - \alpha r} } \right)^{\beta } \left( {1 - e^{ - \alpha r} } \right)^{{\eta + \frac{1}{2}}} P_{3}^{{\left( {2\beta ,2\eta } \right)}} \left( {1 - 2e^{ - \alpha r} } \right)$$

(43)

Thermomagnetic energy spectra of 2-dimensional Schrodinger equation under the influence Aharanov-Bohm (AB) flux and external magnetic field using super symmetric quantum mechanics approach

The supersymmetric approach deals with partner Hamiltonian of the form

$$H_{ \pm } = \frac{{p^{2} }}{2m} + V\left( x \right).$$

(44)

where $p$ is the momentum and $V\left( x \right)$ is the effective potential. The effective potential can be expressed in terms of super potential as

$$V_{eff \pm } \left( x \right) = \phi^{2} \left( x \right) \pm \phi^{\prime}\left( x \right)$$

(45)

The ground state energy is obtained as

$$\phi^{ - 1} \left( x \right) = Ce^{ - N}$$

(46)

where $N$ is the normalization constant which for a very simple case can be determined using the expression

$$N\left( x \right) = \int\limits_{{x_{0} }}^{x} {\phi \left( r \right)} dr.$$

(47)

However, the super potential satisfies the shape invariance condition

$$V_{ + } \left( {a_{0} ,x} \right) = V_{ - } \left( {a_{1} ,x} \right) + R\left( {a_{1} } \right)$$

(48)

where $a_{1}$ is a new set of parameter determines from the old set $a_{0}$ through the mapping $f:a_{0} \to a_{1} = f\left( {a_{0} } \right)$.

The total supersymmetric energy is defined as

$$E_{n} = \sum\limits_{k = 1}^{n} {R(a_{k} )} .$$

(49)

While higher order state solutions are obtained through the expression

$$\phi_{n}^{ - 1} \left( {a_{0} ,x} \right) = \mathop \Pi \limits_{k = 0}^{n - 1} \left( {\frac{{A^{ + } \left( {a_{k} } \right)}}{{\left( {E_{k} - E_{k} } \right)^{\frac{1}{2}} }}} \right)\phi_{0}^{ - 1} \left( {a_{n} ,x} \right).$$

(50)

where $A^{ + } \left( {a_{k} } \right)$ is a raising ladder operator expressed as

$$A^{ + } \left( {a_{k} } \right) = - \frac{\partial }{\partial x} + \phi \left( {a_{k} ,x} \right)$$

(51)

Also, the Schrodinger equation under super symmetric quantum mechanics approach is arranged in the form

$$- \frac{{d^{2} R_{nm} (r)}}{{dr^{2} }} + V_{eff} (\xi ,\omega_{c} ,r)R_{nl} (r) = \tilde{E}_{nm} R_{nm} (r).$$

(52)

With the help of approximation to centrifugal term, Eq. (24) can be re- arranged as follows

$$\begin{aligned} & - \frac{{d^{2} R_{nm} (r)}}{{dr^{2} }} + \left[ \begin{gathered} \frac{{2\mu D\sigma_{0}^{2} \left( {1 + e^{ - \alpha r} } \right)^{2} }}{{\hbar^{2} \left( {1 - e^{ - \alpha r} } \right)^{2} }} - \frac{{4\mu D\sigma_{0} \left( {1 + e^{ - \alpha r} } \right)}}{{\hbar^{2} \left( {1 - e^{ - \alpha r} } \right)}} - \frac{2\mu }{{\hbar^{2} }}\left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) \hfill \\ + \frac{{2\mu \alpha \omega_{c} \left( {m + \xi } \right)e^{ - \alpha r} }}{{\hbar \left( {1 - e^{ - \alpha r} } \right)^{2} }} + \left( {\frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }}} \right)\frac{{e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} + \left( {\frac{{\alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)}}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) \hfill \\ \end{gathered} \right]\\ &\quad R_{nm} (r) = \left( {\frac{{2\mu E_{nm} }}{{\hbar^{2} }} - \frac{2\mu D}{{\hbar^{2} }}} \right)R_{nm} (r). \end{aligned}$$

(53)

Equation (53) can then be compared to Eq. (52) such that

$$\begin{aligned} V_{eff} (\xi ,\omega_{c} ,r) & = \frac{{2\mu D\sigma_{0}^{2} \left( {1 + e^{ - \alpha r} } \right)^{2} }}{{\hbar^{2} \left( {1 - e^{ - \alpha r} } \right)^{2} }} - \frac{{4\mu D\sigma_{0} \left( {1 + e^{ - \alpha r} } \right)}}{{\hbar^{2} \left( {1 - e^{ - \alpha r} } \right)}} - \frac{2\mu }{{\hbar^{2} }}\left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) \\ & \;\;\; + \frac{{2\mu \alpha \omega_{c} \left( {m + \xi } \right)e^{ - \alpha r} }}{{\hbar \left( {1 - e^{ - \alpha r} } \right)^{2} }} + \left( {\frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }}} \right)\frac{{e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} + \left( {\frac{{\alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)}}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) \\ \tilde{E}_{nm} & = \left( {\frac{{2\mu E_{nm} }}{{\hbar^{2} }} - \frac{2\mu D}{{\hbar^{2} }}} \right) \\ \end{aligned}$$

(54)

The proposed super potential that is suitable for the effective potential is given as

$$\omega (r) = f - \frac{{ge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)}} \Rightarrow \omega^{\prime}(r) = \frac{{\alpha ge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}$$

(55)

The supersymmetric partner potential can be obtained as follows :

$$\begin{gathered} V_{eff}^{ + } (\xi ,\omega_{c} ,r) = \omega^{2} (r) + \omega^{\prime}(r) = f^{2} - \frac{{2fge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)}} + \frac{{g^{2} e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} + \frac{{\alpha ge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} \hfill \\ V_{eff}^{ - } (\xi ,\omega_{c} ,r) = \omega^{2} (r) - \omega^{\prime}(r) = f^{2} - \frac{{2fge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)}} + \frac{{g^{2} e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} - \frac{{\alpha ge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} \hfill \\ \end{gathered}$$

(56)

Equation (56) obeys shape invariance condition

$$V_{ + } \left( {r,a_{j} } \right) = V_{ - } \left( {r,a_{j + 1} } \right) + R\left( {a_{j} } \right)$$

(57)

The ground state energy can be obtained by solving the associated Riccati equation. Hence, the supersymmetric partner potential of Eq. (56) has a null ground state energy which implies that

$$\begin{aligned} V_{ - } \left( r \right) & = V_{eff} \left( {r,\xi ,\omega_{c} } \right) - \overline{E}_{0k} \\ & \Rightarrow f^{2} - \frac{{2fge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)}} + \frac{{g^{2} e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} - \frac{{\alpha ge^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} \\ & = \frac{{2\mu D\sigma_{0}^{2} \left( {1 + e^{ - \alpha r} } \right)^{2} }}{{\hbar^{2} \left( {1 - e^{ - \alpha r} } \right)^{2} }} - \frac{{4\mu D\sigma_{0} \left( {1 + e^{ - \alpha r} } \right)}}{{\hbar^{2} \left( {1 - e^{ - \alpha r} } \right)}} - \frac{2\mu }{{\hbar^{2} }}\left( {\frac{{c_{1} e^{ - 2\alpha r} + c_{2} e^{ - \alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) \\ & \;\;\; + \frac{{2\mu \alpha \omega_{c} \left( {m + \xi } \right)e^{ - \alpha r} }}{{\hbar \left( {1 - e^{ - \alpha r} } \right)^{2} }} + \left( {\frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }}} \right)\frac{{e^{ - 2\alpha r} }}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }} + \left( {\frac{{\alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)}}{{\left( {1 - e^{ - \alpha r} } \right)^{2} }}} \right) - \overline{E}_{0k} . \\ \end{aligned}$$

(58)

Expanding Eq. (58) in ascending powers of exponent gives rise to three simultaneous equations of the form

$$f^{2} + 2fg + g^{2} = \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} + \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} - \frac{{2\mu c_{1} }}{{\hbar^{2} }} + \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }} - \overline{E}_{0,k} .$$

(59)

$$- f^{2} - 2fg - \alpha g^{2} = \frac{{4\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} - \frac{{2\mu c_{2} }}{{\hbar^{2} }} + \frac{{2\mu \alpha \omega_{c} \left( {m + \xi } \right)}}{\hbar } + 2\overline{E}_{0,k}$$

(60)

$$\begin{aligned} f^{2} & = \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} - \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} + \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) - \overline{E}_{0,k} \\ \Rightarrow \overline{E}_{0,k} & = \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} - \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} + \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) - f^{2} \\ \end{aligned}$$

(61)

The ground state energy is calculated using Eq. (61). Substituting Eq. (61) into (60) as well as Eq. (61) into (59) and simplifying gives the following equation

$$- 2fg - \alpha g = \frac{{8\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} }} - \frac{{2\mu c_{2} }}{{\hbar^{2} }} + \frac{{2\mu \alpha \omega_{c} \left( {m + \xi } \right)}}{\hbar } + 2\alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)$$

(62)

$$2fg + g^{2} = \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} }} - \frac{{2\mu c_{1} }}{{\hbar^{2} }} + \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }} - \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)$$

(63)

Adding Eq. (62) to Eq. (63) and simplifying gives a quadratic equation of the form:

$$g^{2} - \alpha g - \left( {\frac{{8\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} + \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }} - \frac{{2\mu c_{1} }}{{\hbar^{2} }} - \frac{{2\mu c_{2} }}{{\hbar^{2} }} + \frac{{2\mu \alpha \omega_{c} \left( {m + \xi } \right)}}{\hbar } + \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)} \right) = 0$$

(64)

The solution to Eq. (64) is

$$g = \alpha \left[ {\frac{1}{2} \pm \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{\hbar^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{\hbar^{2} \alpha^{2} }} + \frac{{8\mu \omega_{c} \left( {m + \xi } \right)}}{\hbar \alpha } + 4\left( {m + \xi } \right)^{2} } } \right]$$

(65)

Using Eq. (63), the constant $f$ can be evaluated as

$$f = \frac{1}{2}\left\{ { - g + \frac{{\left[ {\frac{{8\mu D\sigma_{0} }}{{\hbar^{2} }} - \frac{{2\mu c_{1} }}{{\hbar^{2} }} + \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} }} - \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right)} \right]}}{g}} \right\}$$

(66)

The excited state energy is calculated using shape invariance condition

$$V_{ - } \left( {r,\xi ,\omega_{c} ,g} \right) = V_{eff} \left( {r,\xi ,\omega_{c} } \right) - \overline{E}_{0,k} .$$

(67)

If $g = g_{0} \begin{array}{*{20}c} , & {g_{1} = g_{0} + 1\begin{array}{*{20}c} , & {g_{n} = g_{0} + \alpha n} \\ \end{array} } \\ \end{array}$. Then using Eq. (67), then, the shape invariance condition equation become

$$\begin{aligned} V_{ + } \left( {r,\xi ,\omega_{c} ,g_{0} } \right) & = V_{eff} (r,\xi ,\omega_{c} ) - \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} + \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} - \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) \\ & \;\;\; + \frac{1}{4}\left\{ {g_{0} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{0} }}} \right\}^{2} \\ \end{aligned}$$

(68)

$$\begin{aligned} V_{ - } \left( {r,\xi ,\omega_{c} ,g_{1} } \right) & = V_{eff} (r,\xi ,\omega_{c} ) - \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} + \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} - \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) \\ & \;\;\; + \frac{1}{4}\left\{ {g_{1} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{1} }}} \right\}^{2} \\ \end{aligned}$$

(69)

We can then construct the supersymmetric partner potentials of the form

$$\begin{aligned} R\left( {g_{1} } \right) & = V_{ + } \left( {r,\xi ,\omega_{c} ,g_{0} } \right) - V_{ - } \left( {r,\xi ,\omega_{c} ,g_{1} } \right) \\ & \Rightarrow \frac{1}{4}\left\{ {g_{0} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{0} }}} \right\}^{2} \\ &\quad- \frac{1}{4}\left\{ {g_{1} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{1} }}} \right\}^{2} \\ \end{aligned}$$

(71)

$$\begin{aligned} R\left( {g_{2} } \right) & = V_{ + } \left( {r,\xi ,\omega_{c} ,g_{1} } \right) - V_{ - } \left( {r,\xi ,\omega_{c} ,g_{2} } \right) \\ & \Rightarrow \frac{1}{4}\left\{ {g_{1} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{1} }}} \right\}^{2} \\ & \quad - \frac{1}{4}\left\{ {g_{2} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{2} }}} \right\}^{2} \\ \end{aligned}$$

(72)

$$\begin{aligned} R\left( {g_{3} } \right) & = V_{ + } \left( {r,\xi ,\omega_{c} ,g_{2} } \right) - V_{ - } \left( {r,\xi ,\omega_{c} ,g_{3} } \right) \\ & \Rightarrow \frac{1}{4}\left\{ {g_{2} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{2} }}} \right\}^{2} \\ &\quad- \frac{1}{4}\left\{ {g_{3} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{3} }}} \right\}^{2} \\ \end{aligned}$$

(73)

$$\begin{aligned} R\left( {g_{n} } \right) & = V_{ + } \left( {r,\xi ,\omega_{c} ,g_{n - 1} } \right) - V_{ - } \left( {r,\xi ,\omega_{c} ,g_{n} } \right) \\ & \Rightarrow \frac{1}{4}\left\{ {g_{n - 1} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{n - 1} }}} \right\}^{2} \\ &\quad- \frac{1}{4}\left\{ {g_{n} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{n} }}} \right\}^{2} \\ \end{aligned}$$

(74)

Recall that $g = g_{0} \begin{array}{*{20}c} , & {g_{n} = g_{0} + \alpha n} \\ \end{array} = g + \alpha n$. Using Eq. (74), the higher order supersymmetric energy can be evaluated as

$$\overline{E}_{nk} = \frac{1}{4}\left\{ {g_{0} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{0} }}} \right\}^{2} - \frac{1}{4}\left\{ {g_{n} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{n} }}} \right\}^{2}$$

(75)

Meanwhile, the total energy is the ground state energy plus higher order supesymmetric energy

$$\tilde{E}_{nm} = \sum\limits_{k = 1}^{n} {R(a_{k} )} = \overline{E}_{0k} + \overline{E}_{nk}$$

(76)

Substituting Eqs. (61) and (75) together with Eq. (66) into Eq. (76) and simplifying gives the total energy as

$$\tilde{E}_{nm} = \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} - \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} + \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) - \frac{1}{4}\left\{ {g_{n} \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{g_{n} }}} \right\}^{2}$$

(77)

Using the supersymmetric mapping $g_{n} : \to g + \alpha n$ with the total energy expressed as $\tilde{E}_{nm} = \left( {\frac{{2\mu E_{nm} }}{{\hbar^{2} }} - \frac{2\mu D}{{\hbar^{2} }}} \right)$, Eq. (77) now become

$$\begin{aligned} \left( {\frac{{2\mu E_{nm} }}{{\hbar^{2} }} - \frac{2\mu D}{{\hbar^{2} }}} \right) & = \frac{{2\mu D\sigma_{0}^{2} }}{{\hbar^{2} }} - \frac{{4\mu D\sigma_{0} }}{{\hbar^{2} }} + \alpha^{2} \left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) \\ & \;\; - \frac{1}{4}\left\{ {\left( {g + \alpha n} \right) \pm \frac{{\alpha^{2} \left[ {\frac{{2\mu c_{1} }}{{\hbar^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{\hbar^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4}} \right]}}{{\left( {g + \alpha n} \right)}}} \right\}^{2} \\ \end{aligned}$$

(78)

Substituting the value of $g$ from Eq. (65) into Eq. (78) and factorizing gives

$$\begin{aligned} E_{nm} & = \frac{{h^{2} \alpha^{2} }}{2\mu }\left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) + D + D\sigma_{0}^{2} - 2D\sigma_{0} \\ & \;\;\; - \frac{{h^{2} \alpha^{2} }}{8\mu }\left\{ {\frac{\begin{gathered} \left[ {n + \frac{1}{2} \pm \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]^{2} \hfill \\ + \frac{{2\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4} \hfill \\ \end{gathered} }{{\left[ {n + \frac{1}{2} \pm \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right]}}} \right\} \\ \end{aligned}$$

(79)

With a high level of analytical mathematical accuracy, it can be shown that the energy eigen equation obtained through (NUFA) as shown in Eq. (34) reproduces the exact results obtained through SUSYQM as shown in Eq. (79). This further confirms the accuracy of NUFA method in providing bound state solutions to exponential type potentials. Equation (79) can be presented in a more simplified form as:

$$E_{nm} = Q_{1} - Q_{2} \left[ {\left( {n + \delta } \right) + \frac{{Q_{3} }}{{\left( {n + \delta } \right)}}} \right]^{2}$$

(80)

where

$$\begin{aligned} Q_{1} & = \frac{{h^{2} \alpha^{2} }}{2\mu }\left( {\left( {m + \xi } \right)^{2} - \frac{1}{4}} \right) + D + D\sigma_{0}^{2} - 2D\sigma_{0} \\ Q_{2} & = \frac{{h^{2} \alpha^{2} }}{8\mu } \\ Q_{3} & = \frac{{2\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu D\sigma_{0} }}{{h^{2} \alpha^{2} }} + \left( {m + \xi } \right)^{2} - \frac{1}{4} \\ \delta & = \left[ {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{{32\mu D\sigma_{0}^{2} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{1} }}{{h^{2} \alpha^{2} }} - \frac{{8\mu c_{2} }}{{h^{2} \alpha^{2} }} + \frac{{4\mu^{2} \omega_{c}^{2} }}{{h^{2} \alpha^{2} }} + 4\left( {m + \xi } \right)^{2} + \frac{{8\mu \omega_{c} }}{h\alpha }\left( {m + \xi } \right)} } \right] \\ \end{aligned}$$

(81)

Thermomagnetic properties

The thermodynamic properties of quantum systems can be obtained from the exact partition function given by

$$Z\left( \beta \right) = \sum\limits_{n = 0}^{\lambda } {e^{{ - \beta E_{n} }} }$$

(82)

where, $\lambda$ is an upper bound of the vibrational quantum number obtained from the numerical solution of $\frac{{dE_{n} }}{dn} = 0$$,$ given as $\lambda = - \delta + \sqrt {Q_{3} }$, $\beta = \frac{1}{kT}$ where $k$ and T are Boltzmann constant and absolute temperature respectively. In the classical limit, the summation in Eq. (82) can be replaced with an integral:

$$Z(\beta ) = \int\limits_{0}^{\lambda } {e^{{ - \beta E_{n} }} dn} .$$

(83)

Using Eq. (83), the partition function can be expressed as

$$Z(\beta ) = e^{{\beta \left( {2Q_{2} Q_{3} - Q_{1} } \right)}} \int\limits_{0}^{\lambda } {e^{{\beta \left( {Q_{2} \rho^{2} + \frac{{Q_{2} Q_{3}^{2} }}{{\rho^{2} }}} \right)}} d\rho } ,$$

(84)

where $\rho = n + \delta$.

Using Mathematica 10.0 version, Eq. (84) can be evaluated as

$$\frac{{e^{{2\beta Q_{2} Q_{3} - \beta Q_{1} }} \sqrt \pi \left[ {e^{{2\beta Q_{2} Q_{3} }} erf\left( {\sqrt { - \beta Q_{2} } \lambda + \frac{{Q_{3} \sqrt { - \beta Q_{2} } }}{\lambda }} \right) + e^{{ - 2\beta Q_{2} Q_{3} }} erf\left( {\sqrt { - \beta Q_{2} } \lambda - \frac{{Q_{3} \sqrt { - \beta Q_{2} } }}{\lambda }} \right)} \right]}}{{4\sqrt { - \beta Q_{2} } }}.$$

(85)

Using Eq. (85), other thermo-magnetic properties can be obtained as follows

• (a) Vibrational internal mean energy. The vibrational internal mean energy⁴⁴ is defined as

  $$U\left( {\beta ,B,\Phi_{AB} } \right) = - \frac{\partial }{\partial \beta }InZ\left( {\beta ,B,\Phi_{AB} } \right)$$

  (86)

• (b) Free energy. The vibrational free energy⁴⁴ is evaluated as

  $$F\left( {\beta ,B,\Phi_{AB} } \right) = - \frac{1}{\beta }InZ\left( {\beta ,B,\Phi_{AB} } \right)$$

  (87)

• (c) The magnetization at finite temperature⁴⁴ is given as

  $$M\left( {\beta ,B,\Phi_{AB} } \right) = \frac{1}{\beta }\left( {\frac{1}{{Z\left( {\beta ,B,\Phi_{AB} } \right)}}} \right)\frac{\partial }{\partial B}\left( {\beta ,B,\Phi_{AB} } \right)$$

  (88)

• (d) Magnetization of a system at zero temperature in a state ($n,m$) is defined by⁴⁴ as

  $$M_{nm} \left( {\beta ,\Phi_{AB} } \right) = \frac{{\partial E_{nm} }}{\partial B}$$

  (89)

• (e) Magnetic susceptibility⁴⁴ at finite temperature is given as

  $$\chi_{nm} \left( {\beta ,B,\Phi_{AB} } \right) = \frac{{\partial \left( {\beta ,B,\Phi_{AB} } \right)}}{\partial B}$$

  (90)

• (f) Persistent current

  $$I\left( \beta \right) = - \frac{e}{hc}\frac{{\partial F\left( {\beta ,B,\Phi_{AB} } \right)}}{{\partial \Phi_{AB} }}$$

  (91)

• (g) The entropy⁴⁴ of the thermo-magnetic system is given as

  $$k\ln Z\left( {\beta ,B,\Phi_{AB} } \right) - k\beta \frac{{\partial Z\left( {\beta ,B,\Phi_{AB} } \right)}}{\partial B}$$

  (92)

• (h) Specific heat capacity⁴⁴of the system is given as

  $$C_{s} \left( {\beta ,B,\Phi_{AB} } \right) = k\beta^{2} \frac{{\partial^{2} }}{{\partial \beta^{2} }}\ln Z\left( {\beta ,B,\Phi_{AB} } \right)$$

  (93)

Fisher information entropies

In this section, we shall examine the effects of the Aharanov-Bohm (AB) flux and external magnetic field on Fisher information entropy using the proposed potential. Fisher and other quantum information entropies measure the spread of probability distribution for an allowed quantum mechanical state in a D-dimensional space^57,58,59. Fisher information has a lot of applications, including the characterizing of complex signals of quantum mechanical systems, derivation of the equation of motion⁶⁰, investigating the behavior of stock market patterns⁶¹ as well as providing useful information about localization of quantum mechanical particles in a bounded potential well⁶². Fisher entropy expressed in terms of both momentum and position spaces^63,64 are:

$$I\left( \rho \right) = 4\int\limits_{0}^{\infty } {\left| {\nabla \psi \left( {r,\omega_{c} ,\xi } \right)} \right|^{2} } dr$$

(94)

$$I\left( \gamma \right) = 4\int\limits_{0}^{\infty } {\left| {\nabla \psi \left( {p,\omega_{c} ,\xi } \right)} \right|^{2} } dp$$

(95)

For 2-Dimensional Schrodinger wave equation, the Fisher uncertainty product satisfies the inequality⁶⁵

$$I\left( \rho \right)I\left( \gamma \right) \ge 16\left( {\left| m \right| + 1} \right)^{2}$$

(96)

For a two dimensional problem, the momentum space wave function is expressed as

$$\psi_{nm} \left( {p,\phi_{p} ,\omega_{c} ,\xi } \right) = \frac{1}{{\left( {2\pi } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}\int\limits_{0}^{\infty } {\int\limits_{0}^{2\pi } {\frac{{R_{nm} \left( {r,\omega_{c} ,\xi } \right)}}{\sqrt r }} e^{{i\left[ {m\phi_{r} - pr\cos \left( {\phi_{r} - \phi_{p} } \right)} \right]}} rdrd\phi_{r} }$$

(97)

where the solution of the angular part is expressed interms of Bessel function as

$$\int\limits_{0}^{2\pi } {e^{{i\left[ {m\phi_{r} - pr\cos \left( {\phi_{r} - \phi_{p} } \right)} \right]}} } d\phi_{r} = \left( { - 1} \right)^{m} 2\pi J_{m} \left( {pr} \right)e^{{im\phi_{p} }}$$

(98)

and $J_{\left| m \right|} \left( {pr} \right)$ is the Bessel function of order $m$.

The momentum space wave function is either obtained through a Fourier transform or through expectation value expression. For the purpose of this work, we shall be considering the simplest case where the magnetic quantum spin $m = 0$. Therefore, for momentum space wave function in 2D for $m = 0$, is calculated using expectation value equation of the form

$$I\left( \gamma \right) = 4\int\limits_{0}^{\infty } {\left| {\nabla \psi \left( {r,\omega_{c} ,\xi } \right)} \right|^{2} } r^{2} dr$$

(99)

Analytical evaluation of Fisher information entropies for some quantum state

The normalized ground state wave function under the influenced of Aharanov-Bohm flux and external magnetic field is presented in Eq. (39). The gradient of the normalized ground state wave function is given as

$$\nabla \psi_{0} \left( {r,\omega_{c} ,\xi } \right) = - \frac{1}{2}\left( {e^{ - \alpha r} } \right)^{\beta } \left( {1 - e^{ - \alpha r} } \right)^{{ - \frac{1}{2} + \eta }} \alpha \left( {2\left( {e^{ - \alpha r} - 1} \right)\beta - 2\eta - 1} \right)\sqrt {\frac{{\alpha \Gamma \left( {2\beta + 2\eta + 2} \right)}}{{\Gamma \left( {2\beta } \right)\Gamma \left( {2 + 2\eta } \right)}}}$$

(100)

Substituting Eq. (100) into Eq. (94) gives the fisher information in position space as

$$I(p)_{n = 0} = \frac{{\alpha^{2} \left( {1 + 2\eta } \right)\Gamma \left( {1 + 2\beta } \right)\Gamma \left( {2\beta + 2\eta + 2} \right)}}{{\Gamma \left( {2\beta } \right)\Gamma \left( {2 + 2\eta } \right)\Gamma \left( {2\beta + 2\eta + 1} \right)}}$$

(101)

Using Eq. (99), Fisher information in momentum space expressed in terms of polygamma function by the help of Mathematica 10.0 version is given as

$$I(\gamma )_{n = 0} = \frac{2}{{\alpha^{2} \left( {1 + 2\eta } \right)}}\left\{ \begin{gathered} \frac{4 + 8\eta }{{\left( {1 + 2\beta + 2\eta } \right)^{2} }} - \left( {\frac{{4\left( {1 + 2\eta } \right)\psi^{\left( 0 \right)} \left( {0,2\beta } \right)}}{{\left( {1 + 2\beta + 2\eta } \right)}}} \right) + \left( {\frac{{\left( {4 + 8\eta } \right)\psi^{\left( 0 \right)} \left( {0,1 + 2\beta + 2\eta } \right)}}{{\left( {1 + 2\beta + 2\eta } \right)}}} \right) \hfill \\ + \left( {2 + 4\eta } \right)\psi^{\left( 0 \right)} \left( {0,2\beta } \right) - 4\left( {1 + 2\eta } \right)\psi^{\left( 0 \right)} \left( {0,2\beta } \right)\psi^{\left( 0 \right)} \left( {0,1 + 2\beta + 2\eta } \right) \hfill \\ + 2\left( {1 + 2\eta } \right)\psi^{\left( 0 \right)} \left( {0,1 + 2\beta + 2\eta } \right)^{2} \hfill \\ + \left( {2 + 4\eta } \right)\psi^{\left( 0 \right)} \left( {1,2\beta } \right) - 2\left( {1 + 2\eta } \right)\psi^{\left( 0 \right)} \left( {1,1 + 2\beta + 2\eta } \right) \hfill \\ \end{gathered} \right\}$$

(102)

Using the same procedure, we can obtain Fisher information for other quantum states. For $n = 1$, the Fisher information for both position and momentum spaces are given as

$$I(p)_{n = 1} = \frac{{ - 2\alpha^{2} \beta \left( {1 + 2\eta } \right)\Gamma \left( {3 + 2\beta + 2\eta } \right)\Gamma \left( {1 + 6\eta } \right)\Gamma \left( {2\eta } \right)}}{{\Gamma \left( {3 + 2\eta } \right)\Gamma \left( {2 + 2\eta } \right)}}$$

(103)

$$I(\gamma )_{n = 1} = 4\left[ {\frac{{\Xi_{0} + \Xi_{1} + \Xi_{2} + \Xi_{3} + \Xi_{4} + \Xi_{5} + \Xi_{6} + \Xi_{7} }}{{\alpha^{2} \left( {1 + 2\beta } \right)\left( {1 + 2\eta } \right)\left( {3 + 2\eta } \right)\left( {3 + 2\beta + 2\eta } \right)^{2} }}} \right]$$

(104)

where

$$\begin{aligned} \Xi_{0} & = 86 + 94\beta - 64\beta^{2} + 42\left( {47 + 56\beta } \right)\eta + 4\left( {43 + 30\beta } \right)\eta^{2} + 16\left( {5 + \beta } \right)\eta^{3} + 16\eta^{4} \\ \Xi_{1} & = \frac{{\beta \left( {1 + 2\beta } \right)^{2} }}{{\left( {1 + \beta + \eta } \right)^{2} }} + \frac{{\left( {1 + 2\beta } \right)\left( {2\beta - 1} \right)\left( {1 + 3\beta } \right)}}{{\left( {1 + \beta + \eta } \right)}} + \frac{{16\left( {\beta - 1} \right)\beta \left( {1 + 2\beta } \right)}}{{\left( {1 + 2\beta + 2\eta } \right)^{2} }} + \frac{{8 + 8\beta \left( {14\beta^{2} - 5} \right)}}{{\left( {1 + 2\beta + 2\eta } \right)}} \\ \Xi_{2} & = \left( {1 + 2\beta } \right)\left( {3 + 2\beta + 2\eta } \right)^{2} \left( {3 + 8\eta + 4\eta^{2} } \right)\psi^{\left( 0 \right)} \left( {0,2\beta } \right)^{2} \\ \Xi_{3} & = \left\{ {\frac{{2\left( {1 + 2\eta } \right)\left( {3 + 2\beta + 2\eta } \right)\left[ \begin{gathered} \beta^{2} \left( {44 + 40\eta } \right) + \beta \left( {3 + 2\eta } \right)\left( {29 + 56\eta + 28\eta^{2} } \right) \hfill \\ + \left( {1 + \eta } \right)2\left( {3 + 2\eta } \right)\left( {7 + 12\eta + 4\eta^{2} } \right) + 4\beta^{2} \left( {28 + 49\eta + 22\eta^{2} } \right)\psi^{\left( 0 \right)} \left( {0,1 + 2\beta + 2\eta } \right) \hfill \\ \end{gathered} \right]}}{{\left( {1 + \beta + \eta } \right)\left( {1 + 2\beta + 2\eta } \right)}}} \right\} \\ \Xi_{4} & = \left( {1 + 2\beta } \right)\left( {3 + 2\beta + 2\eta } \right)^{2} \left( {3 + 8\eta + 4\eta^{2} } \right)\psi^{\left( 0 \right)} \left( {0,1 + 2\beta + 2\eta } \right)^{2} \\ \Xi_{5} & = \left\{ \begin{gathered} \left( {1 + \beta + \eta } \right)\left( {1 + 2\beta + 2\eta } \right)\left( {2 + 4\eta } \right)\left( {3 + 2\beta + 2\eta } \right)\psi^{\left( 0 \right)} \left( {0,2\beta } \right)\beta^{2} \left( {44 + 40\eta } \right) \hfill \\ + \beta \left( {3 + 2\eta } \right)\left( {29 + 56\eta + \eta^{2} } \right) \hfill \\ \end{gathered} \right\}^{ - 1} \\ \Xi_{6} & = \left( {1 + \eta } \right)\left( {3 + 2\eta } \right)\left( {7 + 12\eta + 4\eta^{2} } \right) + 4\beta^{2} \left( {28 + 49\eta + 22\eta^{2} } \right) \\ & \;\;\; + \left( {1 + 2\beta } \right)\left( {1 + \beta + \eta } \right)\left( {3 + 2\eta } \right)\left( {1 + 2\beta + 2\eta } \right)\left( {3 + 2\beta + 2\eta } \right)\psi^{\left( 0 \right)} \left( {0,1 + 2\beta + 2\eta } \right) \\ \Xi_{7} & = \left( {1 + 2\beta } \right)\left( {3 + 2\beta + 2\eta } \right)^{2} \left( {3 + 8\eta + 4\eta^{2} } \right)\psi^{\left( 0 \right)} \left( {1,2\beta } \right) \\ & \;\;\; + \left( {2\beta - 1} \right)\left( {3 + 2\beta + 2\eta } \right)^{2} \left( {3 + 8\eta + 4\eta^{2} } \right)\psi^{\left( 0 \right)} \left( {1,1 + 2\beta + 2\eta } \right) \\ \end{aligned}$$

(105)

where the polygamma function is generally expressed as

$$\psi^{\left( 0 \right)} (x) = \left( {\ln \Gamma \left( x \right)} \right)^{\prime } = \frac{{\int\limits_{0}^{\infty } {t^{x - 1} } e^{ - t} \ln tdt}}{{\int\limits_{0}^{\infty } {t^{x - 1} } e^{ - t} dt}}$$

(106)

Results and discussion

Figures 1a–d are the plots of variation of thermomagnetic energy spectra against the screening parameter in the absence of AB and magnetic field, the presence of the only magnetic field, the presence of only AB field and the presence of both magnetic and AB fields, respectively. In Fig. 1a–d, the bound state energy spectral diagrams all increases monotonically with increasing values of the screening parameter ($\alpha$) in such a unique and quantized manner.

Figure 1

Variation of thermomagnetic energy spectra against the screening parameter in (a) the absence of AB and Magnetic field, (b) the presence of only magnetic field; (c) the presence of only AB field and (d) the presence of both magnetic and AB field.

Figure 2a, b are the variation of wave function plot against the radial distance in the absence of both AB and magnetic field and the variation of probability density plot against the radial distance in absence of both AB and magnetic field, respectively. In Fig. 2a, the wave function showcases intertwining multiple sinusoidal curves representing the different quantum states. In Fig. 2b, the probability density plots in the absence of AB and magnetic field show a normal distribution curve with multiple peaks, each depicting a different quantum state. It is interesting to note that in Fig. 2a, the ground state has the lowest peak, while the highest state ($n = 3$) has the highest peak. Figure 2b agreed excellently with the theoretical and experimental descriptions of probability density. It is expected that in an ideal condition, the peak of the probability density plot should increase as the quantum state increases. This is only possible because in Fig. 2a and b, the wave function and probability density plots are carried out in the absence of AB and magnetic field respectively.

Figure 2

(a) The variation of wave function plot against the radial distance in the absence of both AB and magnetic field. (b) The variation of probability density plot against the radial distance in absence of both AB and magnetic field.

Figure 3a, b are the variations of the wave function and the probability density plots against the radial distance in the presence of magnetic field. Figure 3a shows a periodic and sinusoidal wave function similar to Fig. 2a. However, in Fig. 3b, there is distortion in the probability distribution curves because of the presence of magnetic field. The presence of the magnetic field does not allow uniform distribution of probability density plots in increasing order of the quantum state whose highest peak supposed to occur at ($n = 3$). However, in Fig. 3b, $n = 2$ has the highest peak, followed by $n = 0$ before $n = 3$. The disorderliness, ambiguity and distortions in the peaks clearly show the effect of magnetic field.

Figure 3

(a) The variation of wave function plot against the radial distance in the presence of magnetic field. (b) The variation of probability density plot against the radial distance in the presence of magnetic field.

Figure 4a, b are the variation of the wave function and probability density plots against the radial distance in the presence of AB field, respectively. Figure 4a and b has similar explanation to Fig. 3a and b when the distortions to the probability density plot are affected by the presence of Aharonov-Bohm flux field.

Figure 4

(a) The variation of wave function plot against the radial distance in the presence of AB field. (b) The variation of probability density plot against the radial distance in the presence of AB field.

Figure 5a, b show how the wave function and probability density varied with radial distance in the presence of both magnetic and AB fields . Under the influence of AB and magnetic fields, the wave function in Fig. 5a is sinusoidal and periodic. .However, in Fig. 5b, something fascinating occurs. The peaks of probability density plot for quantum state ($n = 1$) are almost the same as $n = 2$, i.e., the combined effect of AB and magnetic effect establish quantum state equivalence.

Figure 5

(a) The variation of wave function plot against the radial distance in the presence of both magnetic and AB field. (b) The variation of probability density plot against the radial distance in the presence of both magnetic and AB field.

Figure 6a–d are the plot of partition function against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of partition function against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of partition function against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of partition function against the maximum vibrational quantum number ($\lambda$)for fixed value of $\omega_{c}$ and $\xi$ but for different values of inverse temperature parameter ($\beta$), respectively. In Fig. 6a, the partition function starts from the negative y-axis an increase exponentially with increasing value of the magnetic field. The same explanation occurs in Fig. 6d where the partition function increases exponentially with an increase in maximum vibrational quantum number.

Figure 6

(a) Plot of partition function against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of partition function against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of partition function against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of partition function against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

In Fig. 6b, the partition function rises monotonically with unique spacing before reaching a peak value with local maximum turning point at $\xi = 40$. In Fig. 6c, the partition function increases monotonically with an increase in inverse temperature parameter.

Figure 7a–d are the plot of vibrational mean energy against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of vibrational mean energy against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of vibrational mean energy against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of vibrational mean energy against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively.

Figure 7

(a) Plot of vibrational mean energy against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of vibrational mean energy against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of vibrational mean energy against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of vibrational mean energy against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

In Fig. 7a, the vibrational mean energy showcase a parabolic curve which increases with an increase in magnetic field. In Fig. 7b, the vibrational mean energy increases monotonically before converging at $\xi = 6$ with increase in AB flux. Also, the vibrational mean energy increases uniquely from the origin with quantized spacing, in an increasing value of inverse temperature parameter as shown in Fig. 7c. Correspondingly, the vibrational mean energy also increase with increasing value of maximum vibrational quantum number as presented in Fig. 7d.

Figure 8a–d are the plot of vibrational heat capacity against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of vibrational heat capacity against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of vibrational heat capacity against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of vibrational heat capacity against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively.

Figure 8

(a) Plot of vibrational heat capacity against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of vibrational heat capacity against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of vibrational heat capacity against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of vibrational heat capacity against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

In Fig. 8a, the vibrational heat capacity increases monotonically with increase in magnetic field. In Fig. 8b, the vibrational heat capacity shows symmetrical curves with common converged maximum point at $\xi = 45$ . This maximum point divides the curves into equal half both in a decreasing and increasing value of $\xi$. The physical interpretation of Fig. 8b is that heat capacity from the concept of molecular vibration relates to the kinetic energy of the molecules of the system. So, the Fig. 8b completely shows that with the influence of Aharanov-Bohm flux field, the kinetic energy of the molecules of the system remains constant during molecular vibrations. This explains why there is a symmetrical curves both at the left and right hand side of the thermomagnetic plot. In Fig. 8c, the vibrational heat capacity is a parabolic curve that concaves upward with minimum turning point at $\beta = 0.0005\;\;{\text{K}}^{ - 1}$ before rising to various local maximum turning points in increasing value of $\beta$, before converging at $\beta = 0.004\;\;{\text{K}}^{ - 1}$. In Fig. 8d, the specific heat capacity increases asymmetrically to various unique maximum point before converging at $\lambda = 1000$ with increasing value of maximum vibrational quantum number.

Figure 9a–d are plot of vibrational entropy against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of vibrational entropy against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of vibrational entropy against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of vibrational entropy against the maximum vibrational quantum number ($\lambda$), for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively. In Fig. 9a and d, the vibrational entropy increases exponentially with an increase in magnetic field and maximum vibrational quantum number respectively. In Fig. 9b, the vibrational entropy rises to the peak with maximum turning point at $\xi = 35$ before slopping in a divergence manner with distinct spacing between the spectral curves. In Fig. 9c, the vibrational entropy increases exponentially with an increase in inverse temperature parameter.

Figure 9

(a) Plot of vibrational entropy against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of vibrational entropy against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of vibrational entropy against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of vibrational entropy against the maximum vibrational quantum number ($\lambda$), for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

Figure 10a–d are plot of vibrational Free energy against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of vibrational Free energy against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of vibrational Free energy against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of vibrational Free energy against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively. Figure 10a–d has similar explanation to Fig. 9a–d.

Figure 10

(a) Plot of vibrational Free energy against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of vibrational Free energy against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of vibrational Free energy against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of vibrational Free energy against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

Figure 11a–d. are plot of magnetization against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of magnetization against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of magnetization against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of magnetization against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively. In Fig. 11a, c and d, the magnetization increases exponentially with an increase in $\omega_{c}$, $\beta$ and $\lambda$ respectively. However, in Fig. 11b the influence of AB field produces normal distribution curves with distinct peaks corresponding to the values of inverse temperature parameter ($\beta$).

Figure 11

(a) Plot of magnetization against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of magnetization against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of magnetization against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of magnetization against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

Figure 12a–d are plot of magnetic susceptibility against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$, plot of magnetic susceptibility against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), plot of magnetic susceptibility against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of magnetic susceptibility against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively. In Fig. 12a, the magnetic susceptibility increases monotonically from zero into diverging curves. In Fig. 12b, the magnetic susceptibility produces sinusoidal curves with discontinuity at $\xi = 50$. In Fig. 12c, the magnetic susceptibility rises to attain various local maximum point at precisely $\beta = 0.125\;{\text{K}}^{ - 1}$. Also, in Fig. 12d, the magnetic susceptibility increases exponentially with an increase in maximum vibrational quantum number.

Figure 12

(a) Plot of magnetic susceptibility against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of magnetic susceptibility against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of magnetic susceptibility against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of magnetic susceptibility against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

Figure 13a–d are plot of persistent current against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$ , plot of persistent current against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$), Fig. 13c plot of persistent current against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$) and plot of persistent current against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$) respectively. In Fig. 13a and d, the persistent current increases asymptotically from the origin with increasing value of magnetic field and maximum upper bound vibrational quantum number. In Fig. 13b, the persistent current rises from the origin to exhibits various maximum points before concaving upward with unique minimum points with maximum at $\xi = 45$. In Fig. 13c, the persistent current increases from the vertical axis in a quantized form before diverging into various spectral curves with increasing value of $\beta$.

Figure 13

(a) Plot of persistent current against magnetic flux $\left( {\omega_{c} } \right)$ for different values of inverse temperature parameter $(\beta )$. (b) Plot of persistent current against AB flux $\left( \xi \right)$ for different values of inverse temperature parameter ($\beta$). (c) Plot of persistent current against inverse temperature parameter ($\beta$) for fixed value of $\omega_{c}$ and $\xi$ but for different values of maximum vibrational quantum number ($\lambda$). (d) Plot of persistent current against the maximum vibrational quantum number ($\lambda$) for fixed value of $\omega_{c}$ and $\xi$ but for different values inverse temperature parameter ($\beta$).

Figure 14a–c are the plot of position space Fisher entropy against the screening parameter for $n = 0$, the plot of momentum space Fisher entropy against the screening parameter for $n = 0$ and the plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 0$ respectively. In Fig. 14a, the position space Fisher entropy increases linearly with an increase in the screening parameter, while the momentum space and its product increases exponentially with an increase in the screening parameter ($\alpha$) as shown in Fig. 14b and c respectively.

Figure 14

(a) The plot of position space Fisher entropy against the screening parameter for $n = 0$. (b) The plot of momentum space Fisher entropy against the screening parameter for $n = 0$. (c) The plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 0$.

Figure 15a–c are the plot of position space Fisher entropy against the screening parameter for $n = 1$, the plot of momentum space Fisher entropy against the screening parameter for $n = 1$ and the plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 1$ respectively. Figure 15a–c has the same explanation as Fig. 14a–c.

Figure 15

(a) The plot of position space Fisher entropy against the screening parameter for $n = 1$ . (b) The plot of momentum space Fisher entropy against the screening parameter for $n = 1$. (c) The plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 1$.

Figure 16a–c are the plot of position space Fisher entropy against the screening parameter for $n = 2$, plot of momentum space Fisher entropy against the screening parameter for $n = 2$ and the plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 2$ respectively. Figure 16a–c has the same explanation as Fig. 14a–c.

Figure 16

(a) The plot of position space Fisher entropy against the screening parameter for $n = 2$. (b) The plot of momentum space Fisher entropy against the screening parameter for $n = 2$. (c) The plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 2$.

Figure 17a–c are the plot of position space Fisher entropy against the screening parameter for $n = 3$, the plot of momentum space Fisher entropy against the screening parameter for $n = 3$ and the plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 3$ respectively. In Fig. 17a, the position space entropy increases exponentially with an increase in the value of $\alpha$. However, in Fig. 17b and c, there is abnormally which makes the plot to decrease with decreasing value of $\alpha$ with respect to momentum space and its products respectively.

Figure 17

(a) The plot of position space Fisher entropy against the screening parameter for $n = 3$. (b) The plot of momentum space Fisher entropy against the screening parameter for $n = 3$. (c) The plot of product of position and momentum space Fisher entropy against the screening parameter for $n = 3$.

Table 1 is the numerical bound state solution for the proposed potential under the influence of AB and Magnetic field for fixed magnetic quantum number but with varying principal quantum number. In Table 1, it can be observed that when both fields are deactivated, i.e., AB and magnetic fields are zero, the energy spectra degenerate; thus, as the number of quantum states increases, the energy spectra decrease. When only the AB field was applied to the quantum system, it resulted in quasi-degeneracy, and the energy spectra decreased with increasing quantum states. When only the magnetic field is activated, the system produces a similar effect, but this time degeneracy is gradually eliminated. When both fields are activated, the combined effects completely eliminates degeneracy from the quantum system's energy spectra. All computation were carried out using the following constant physical parameters: $c_{1} = c_{2} = 1,\sigma_{0} = 0.5,\hbar = \mu = 1,\alpha = 0.2,c = 1$.

Table 1 Numerical bound state solution for the proposed potential under the influence of AB and Magnetic field.

Tables 2, 3, 4 and 5 are the numerical computation for position, momentum, and products Fisher entropy under the influence of AB and magnetic field for $n = 0$ to $n = 3$, respectively. In these Tables, it is clear that our results obey Heisenberg uncertainty principles in which there is uncertainty in the simultaneous measurement of the position and momentum of quantum mechanical particles. The numerical results also show that as the values for position Fisher entropy increase, the momentum values decrease with an increase in screening parameters. This trend holds for all quantum states in the absence of both magnetic and AB fields, only magnetic fields, only AB fields, and the combined influence of both magnetic and AB fields.

Table 2 Numerical values for position, momentum and products Fisher entropy under the influence of AB and Magnetic field for $n = 0$.

Table 3 Numerical values for position, momentum and products Fisher entropy under the influence of AB and Magnetic field for $n = 1$.

Table 4 Numerical values for position, momentum and products Fisher entropy under the influence of AB and Magnetic field for $n = 2$.

Table 5 Numerical values for position, momentum and products Fisher entropy under the influence of AB and Magnetic field for $n = 3$.

Correspondingly, our numerical results in all quantum states satisfy the 2D local Fisher uncertainty product inequality expressed as ($I\left( \rho \right)I\left( \gamma \right) \ge 16$ as shown in Tables 2, 3, 4 and 5 for all quantum states. All our results clearly show that as the quantum state increases, the values of position increases, while that of momentum and product values decrease. The Fisher product values in all quantum states clearly show the localization of the quantum mechanical particles both in the absence and presence of magnetic and AB fields. Finally, the numerical results from their product indicate that the particle is more localized when the combined effect of AB and magnetic fields on the entropy than the absence of both fields, as shown by ( ($I\left( \rho \right)I\left( \gamma \right) \ge 16$).

Conclusion

In this work, we study analytical solutions, thermomagnetic properties, and its effect on Fisher information entropy with Schioberg plus Manning-Rosen potential using the Nikiforov-Uvarov functional analysis and Supersymmetric quantum mechanics methods. We obtained the energy equation in a closed and compact form both in NUFA and SUSYQM and applied the solution to study partition function and other thermomagnetic properties.

The trend of thermomagnetic plots is in excellent agreement with the work of existing literature. Using the normalized wave function, we obtained the wave function and probability density plots and applied them to study Fisher information entropy in position and momentum spaces. The numerical results show that the combined impact of the magnetic and AB flux fields completely removes the degeneracy on the energy spectra and that increasing the screening parameter increases the position of Fisher entropy while decreasing its momentum, satisfying the 2D local Fisher uncertainty product condition. It also causes both localization and delocalization of quantum particles. Meanwhile, as the quantum state increases under the combined influence of magnetic and AB fields, the results of Fisher entropies and the product increase. Finally, the proposed potential reduces to Schioberg and Manning-Rosen potential as special cases. The wave function and probability density plots were obtained using Maple 10,0 software, while the position and momentum Fisher entropies were obtained using a well-designed Mathematica program.