Investigating Some Diatomic Molecules Bounded by the Two-Dimensional Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

Oluwadare, O. J.; Ilesanmi, E. O.; Abiola, T. O.; Olubosede, O.; Odo, E. A.; Ajibade, S. O.; Oyewumi, K. J.

doi:https://doi.org/10.1155/2022/6565048

Advances in High Energy Physics

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Research Article | Open Access

Volume 2022 | Article ID 6565048 | https://doi.org/10.1155/2022/6565048

Investigating Some Diatomic Molecules Bounded by the Two-Dimensional Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

O. J. Oluwadare,¹E. O. Ilesanmi,¹T. O. Abiola,¹O. Olubosede,¹E. A. Odo,¹S. O. Ajibade,²and K. J. Oyewumi²

Academic Editor: Matteo Beccaria

Received09 May 2022

Revised29 Jun 2022

Accepted26 Jul 2022

Published10 Sept 2022

Abstract

We investigate the nonrelativistic magnetic effect on the energy spectra, expectation values of some quantum mechanical observables, and diamagnetic susceptibility for some diatomic molecules bounded by the isotropic oscillator plus inverse quadratic potential. The energy eigenvalues and normalized wave functions are obtained via the parametric Nikiforov-Uvarov method. The expectation values square of the position , square of the momentum , kinetic energy , and potential energy are obtained by applying the Hellmann-Feynman theorem, and an expression for the diamagnetic susceptibility is also derived. Using the spectroscopic data, the low rotational and low vibrational energy spectra, expectation values, and diamagnetic susceptibility for a set of diatomic molecules (I₂, H₂, CO, and HCl) for arbitrary values, Larmor frequencies are calculated. The computed energy spectra, expectation values, and diamagnetic susceptibility were found to be more influenced by the external magnetic field strength and inverse quadratic potential strength than the vibrational frequencies and the masses of the selected molecules.

1. Introduction

Several studies in quantum mechanics, solid state physics, condensed matter physics, nuclear physics, chemical physics, molecular physics, and other related areas have proven to an outstanding degree that potential models are very important models for stimulating atomic and molecular interaction since it is capable of predicting and describing some behavior of atoms and molecules. It also provides an insight into the understanding of molecular spectra, vibrations and dynamics [1, 2], spin-orbit interaction, relativistic corrections and diamagnetic susceptibility [3, 4], optical properties [5, 6], interband light absorption and interband optical transitions [7, 8], energy and relativistic effects in weakly bound nuclei [9–11], external magnetic fields and/or Aharonov-Bohm flux fields [12–17], interactions between the magnetic and electric fields [18], thermal and/or thermodynamic properties [19–23], spin and pseudospin symmetries [24], and two-body effects [25–28] among others.

One of the important potential models in this regard is the so-called isotropic oscillator plus inverse quadratic potential (IOPIQP) or anharmonic oscillator potential, which has been explored by some authors in both the relativistic and nonrelativistic domains of quantum mechanics [29–32], [33, 34]. An isotropic oscillator (three-dimensional harmonic oscillator) plus inverse quadratic potential may be defined [29–34] as where is the potential strength, represents the mass of the vibrating molecules, and is the angular frequency with which the molecules vibrate in the presence of a magnetic field. Oyewumi [29] employed the hyperradial equation for the isotropic harmonic oscillator plus inverse quadratic potential and presented the normalized hyperradial and hyperangular solutions, and the solutions depend on the dimension as well as the potential parameters. The hidden symmetries and thermodynamic properties for a harmonic oscillator plus inverse square potential have been exposed by Dong et al. [30], while Arda and Sever reported the exact solutions of Schrödinger for this potential within the framework of the Laplace transform technique [31].

In the same vein, Abdelmadjid [32] also studied the exact nonrelativistic quantum spectrum systems for the isotropic harmonic oscillator plus inverse quadratic potential within the formalisms of both Boopp’s shift method and standard perturbation theory in both noncommutativity of the two-dimensional real space and phase (NC-2D: RSP) and presented the exact corrections for the spectrum and the associated noncommutative anisotropic Hamiltonian. Again, due to the unflinching interest, Abdelmadjid [33] looked into the effect of both noncommutativity of the three-dimensional space and phase on the Schrödinger equation with an isotopic harmonic oscillator plus inverse quadratic potential and reported the exact degenerated spectrum associated for noncommutative space and phase.

Furthermore, Dianawati et al. [34] investigated the Schrödinger equation with quantum deformation for a three-dimensional harmonic oscillator plus inverse quadratic potential via the hypergeometric method. The energy spectra which were calculated and visualized by MATLAB R2013a were found to depend on the quantum deformation and quantum number.

It is in the light of the relevance of this potential model that we are motivated to examine the two-dimensional radial Schrödinger equation with the isotropic harmonic oscillator plus inverse quadratic potential in an external magnetic field via the parametric Nikiforov-Uvarov method, obtain the eigensolutions, and discuss the behavior of energy spectra, expectation values of some quantum mechanical observables, and diamagnetic susceptibility for some selected diatomic molecules bounded by this interaction potential model.

The sensitivity of the bounded molecules in an external magnetic field (using arbitrary values of Larmor frequencies), low rotational and vibrational levels, and inverse quadratic potential strength would be adequately investigated. The situation where Larmor frequency implies the absence of an external magnetic field, whereas Larmor frequencies indicate the presence of an external magnetic field. The case of the low vibrational energy level , low rotational energy level , and inverse quadratic potential strengths would be examined.

Other methods that can be used to solve the aforementioned bound state problems include the wave function ansatz method [13], asymptotic iteration method [16], formula method [17], Euler-Maclaurin approximation [23], Laplace transform technique [31, 35], and supersymmetric approach [36], among others. In Section 2, we give a review of the parametric Nikiforov-Uvarov method. Section 3 contains nonrelativistic eigensolutions, expectation values of some quantum mechanical observables, and diamagnetic susceptibility of the isotropic harmonic oscillator plus inverse quadratic potential in an external magnetic field. The results are discussed extensively in Section 4, while the concluding remarks are given in Section 5.

2. Review of the Parametric Nikiforov-Uvarov Method

The parametric Nikiforov-Uvarov method is a straightforward, consistent, and efficient analytical technique for analyzing second-order linear differential equations arising from bound state problems. The choice of this method is due to the fact that it has been proven to an outstanding degree and is well reported to give excellent results in comparison with other methods in the literature [12]. According to Nikiforov and Uvarov [37], the second-order linear differential equation reduces to the generalized equation of hypergeometric type [37, 38]. With an appropriate coordinate transformation , the equation takes the form where and are polynomials, at most in the second degree, and is a first-degree polynomial.

To solve equation (2), one needs to break the wave function into parts as

Therefore, equation (2) reduces to the hypergeometric-type equation: where satisfies the condition , has a negative derivative, and is related to the function by

The parameter is defined by

The energy eigenvalues can be calculated from equation (7). In order to calculate the energy eigenvalues, we need first to determine by using the first derivative of and defining

By solving the resulting quadratic equation for , we obtain the following expression:

Here, is a polynomial with the parameter and the prime denotes the first derivative of the functions and , respectively. The determination of is the essential point in the calculation of It can be obtained by setting the discriminant of the square root to zero [37]; therefore, a general quadratic expression for can be obtained. On substitution of the values of , , , and into equations (7) and (8) and equating (7) and (8), one can evaluate the energy equation for any potential. The wave function in equation (2) satisfies the condition which can be evaluated using the Rodrigues relation. The polynomial solutions are given by where is a normalization constant and the weight function satisfies the following relation:

A more generalized form of equation (2) for any potential may be presented as [39] which satisfies the wave functions of equation (3). By comparing equation (2) with equation (13), we have the following polynomials:

Substituting equation (13) into equation (9), one obtains with the following parametric constants:

According to the Nikiforov-Uvarov method, the discriminant of equation (15) must be set to zero so that the expression for can be quadratically obtained as

Since the negative value of (that is ) gives the bound state solution, we consider

Inserting equation (18) into (15), we have having its first derivative as

Putting equations (14) and (15) into equation (5), one obtains and its first derivative becomes

By applying equation (16) to equation (22), we have

By applying equations (18) and (20) in equation (8), we have

With equations (14) and (22), the parameter in equation (7) becomes

Equating equations (24) and (25), one obtains the bound state energy equation for any potential [37–40] as

Using equations (10)–(12), the wave function parameters can be evaluated as in such a way that the associated wave function in equation (3) becomes where , , , and , are Jacobi polynomials with the following parametric constants:

By considering a special case where , then the associated wave function reduces to the form [39, 40] where is well known as a Laguerre polynomial.

3. Nonrelativistic Eigensolutions of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

For a charged particle moving in a uniform magnetic field, the Hamiltonian of the system may be defined ([16] and the references therein) as where is the mass of the charged particle, is the electronic charge, is the momentum of the charged particle, is the vector potential in the symmetric gauge, is the velocity of light, and is the cylindrical potential representing the potential in equation (1). The Hamiltonian for this system can be evaluated, in the CGS system and in atomic units , as and the Schrödinger equation yields

Since this problem involves two dimensions, therefore, it is sufficient enough to study in polar coordinates within the plane and to employ the following ansatz for the eigenfunction:

Consequently, the radial wave function must satisfy the following radial Schrödinger equation [16, 17]: with the effective potential defined as where , , and symbolize the Larmor frequency, the eigenvalue of the angular momentum, and the energy spectra of the vibrating molecules, respectively. By using the as the isotropic oscillator plus inverse quadratic potential (IOPIQP), the effective potential influenced by an external magnetic field becomes where and represent the mass and angular frequency of the vibrating molecules bounded by the IOPIQP, and the molecular constants for the selected diatomic molecules in this study are displayed in Table 1. Inserting equation (37) into equation (35) and applying a variable , equation (35) can be transformed as

Comparing equation (38) with equation (13), we obtain the following analytical expressions:

Using equations (16) and (29), other values of parametric constants and their analytical values required for the derivation of energy eigenvalues and eigenfunctions are obtained and displayed in Table 2.

Using the analytical values in Table 2 for the parametric constants in equations (26) and (30), the energy eigenvalues and the normalized radial eigenfunctions for the IOPIQP in the presence of an external magnetic field are obtained, respectively, as where , , and is the associated Laguerre polynomial.

3.1. Expectation Values (, , , and ) of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

The Hellmann-Feynman theorem (HFT) is one of the useful techniques for obtaining expectation values of some quantum mechanical observables for any arbitrary values of quantum numbers [41, 42]. Suppose that the Hamiltonian for a particular quantum mechanical system depends on parameter such that and are the eigenvalues and the eigenfunctions, respectively. Therefore, the Hellmann-Feynman theorem (HFT) states that provided that the normalized eigenfunctions are continuous, differentiable with respect to parameter . The effective Hamiltonian of the isotropic oscillator plus inverse quadratic potential in an external magnetic field is given as

To find the expectation value of , we let such that equation (42) becomes

Taking the first derivative of the effective Hamiltonian in equation (43) with respect to vibrational frequency , one obtains

Taking the first derivative of the energy eigenvalues in equation (40) with respect to vibrational frequency , we have

With equations (45) and (46) in equation (44), it is easy for one to evaluate as

In principle and with in equation (44), the expectation values of , , and can be obtained, respectively, as

3.2. Diamagnetic Susceptibility of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

The diamagnetic susceptibility is given [3, 21] as where is the Avogadro number, is the atomic number, is the electronic charge, is the speed of light, and is the effective mass of the vibrating molecules in this study. It has been found that diamagnetism is a fundamental magnetic phenomenon that explains the tendency of electric charges to partially shield the interior of a body from an external magnetic field and that diamagnetic materials possess magnetic effects due to an external field that alters electron motion within the atoms [21]. Using equation (47) in (49) defines the diamagnetic susceptibility for the isotropic oscillator plus inverse quadratic potential in an external magnetic field as where all the symbols have been explained accordingly. The corresponding magnetic moment can be expressed as

4. Results and Discussions

In order to verify the reliability, validity, and consistency of our results, using the molecular constants in Table 1 [21], we present the computed results for the energy spectra, expectation values , , , and , and diamagnetic susceptibility for the selected molecules bounded by the isotropic oscillator plus inverse quadratic potential with varying Larmor frequencies for the case of the low vibrational energy level , low rotational energy level , and inverse quadratic potential strength in Tables 3–11. Figures 1–4 show the variation of expectation values for some quantum mechanical observables as a function of Larmor frequencies for various rotational energy levels with for the selected molecules in Table 2. Figure 5 shows the variation of diamagnetic susceptibility as a function of Larmor frequencies for various rotational energy levels with for the molecules in Table 1. In all the calculations, we have also employed the following recently used conversions: , , and moles. All our results are in their standard units.

It was observed that energy spectra increase with the increase in the magnetic field strength (Larmor frequency), vibrational level, and inverse quadratic potential strength for all the selected molecules. See Tables 3–6. This observation suggests that the energy spectra of the selected molecules would be affected significantly by the external magnetic field. The expectation value which is positive increases with increasing rotational energy level but decreases monotonically towards zero with the increase in magnetic field strengths for all the selected molecules for . See Table 7 and Figure 1. The expectation value which is negative decreases with increasing rotational energy level but increases with the increase in magnetic field strengths and tends to converge at a very high magnetic field strength (Larmor frequencies ) for all the selected molecules for . See Table 8 and Figure 2.

The expectation value which is negative decreases with increasing rotational energy level but increases monotonically with the increase in magnetic field strengths and tends to converge at a very high magnetic field strength (Larmor frequencies ) for all the selected molecules for . See Table 9 and Figure 3. The expectation value which is positive increases with increasing rotational energy level as well as the magnetic field strength. A clear divergence is noticeable at all values of magnetic field strength for all the selected molecules for . See Table 10 and Figure 4.

The diamagnetic susceptibility which is negative increases monotonically with increasing rotational energy level as well as the magnetic field strength and tends to converge at any for all the selected molecules for . See Table 11 and Figure 5.

5. Concluding Remarks

We have studied the two-dimensional radial Schrödinger equation with the isotropic harmonic oscillator plus inverse quadratic potential in an external magnetic field via the parametric Nikiforov-Uvarov method. The energy eigenvalue equation, normalized wave function, expressions for expectation values square of the position , square of the momentum , kinetic energy , and potential energy , and diamagnetic susceptibility for the interaction potential model have been obtained. The computed results for the energy spectra, expectation values , , , and , and diamagnetic susceptibility for some diatomic molecules bounded by the isotropic harmonic oscillator plus inverse quadratic potential for low vibrational and rotational levels are found to have strong dependence on the magnetic field strengths as well as the inverse quadratic potential strength .

The expectation values , , and and diamagnetic susceptibility as a function of Larmor frequencies for all the molecules tend to converge at a very high magnetic field strength . This may signify a case where low vibrational and rotational energy levels cease to have any significant effect on the expectation values of , , and and diamagnetic susceptibility for all the molecules, despite the increasing magnetic field strength. A divergence is noticeable for the expectation value of potential energy which signifies that the rotational and vibration energy levels will continue to have a meaningful effect as long as the magnetic field strength increases. Also, as the magnetic field strength increases, diamagnetic susceptibility increases, which may likely increase the tendency of the magnetic field to shield or alter the vibrational motion of the molecules bounded by the potential model in this study.

Using the results obtained in this work, one can study the thermodynamic properties of the system, Shannon entropy, and Fisher information for the first and second excited states. And this would be our focus in the subsequent research work.

Data Availability

All the data used are included in this paper.

Consent is not applicable.

Conflicts of Interest

The authors declared that there are no competing interests regarding the paper.

Acknowledgments

The authors wish to thank some authors for making their papers available for the development of this paper.

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