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Physical Science International Journal

Thermal and Quantum Properties of Nonlinear and Inverted Harmonic Oscillators in Magnetic Fields: Applications to Polyatomic Molecules

Alalibo Thompson Ngiangia, Peter Onyelukachukwu Nwabuzor

Published Dec 11, 2025• 0 citations
Publisher

Abstract

An analytical study of the radial Schrodinger wave equation with magnetic field, nonlinear and inverted nonlinear isotropic harmonic oscillator potentials was carried out. A modified substitution technique was used to determine the energy eigenvalues of the nonlinear and inverted nonlinear oscillators while the Laplace and computational methods were used to determine the radial wave functions. The determination of the energies’ eigenvalues and the radial wave functions enabled the study to generate tables, plot graphs, and discuss the effect of the magnetic field, angular frequency, and nonlinear term on the thermodynamic functions and systems. Results of the study revealed that the energies’ eigenvalues of the inverted nonlinear oscillator fluctuates increasingly and decreasingly, while that of the nonlinear isotropic oscillator fluctuates decreasingly and degeneracies, completely removed. As the angular frequency is improved, expansion or increase is observed in the nonlinear and the inverse nonlinear cases in the radial wave functions. In the case of increase in the nonlinear term, saturation in the radial wave functions are observed in the inverted and non-inverted cases. For the effect of the increase in the magnetic field potential, initial decrease with saturation was observed in the both cases. The effect of the magnetic field and nonlinear term in both the nonlinear and inverted oscillators on the thermodynamic functions, attained saturation on the polyatomic molecules, while the effects of angular frequency is observed to enhance the thermodynamic functions. Generally, the contraction or saturation experienced with the increase of the magnetic field term is as a result of the small value of the electron charge, and the carbon (iv) oxide molecule is at higher potential than that of the water molecule.

DOI
10.9734/psij/2025/v29i6917

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An analytical study of the radial Schrodinger wave equation with magnetic field, nonlinear and inverted nonlinear isotropic harmonic oscillator potentials was carried out. A modified substitution technique was used to determine the energy eigenvalues of the nonlinear and inverted nonlinear oscillators while the Laplace and computational methods were used to determine the radial wave functions. The determination of the energies’ eigenvalues and the radial wave functions enabled the study to generate tables, plot graphs, and discuss the effect of the magnetic field, angular frequency, and nonlinear term on the thermodynamic functions and systems. Results of the study revealed that the energies’ eigenvalues of the inverted nonlinear oscillator fluctuates increasingly and decreasingly, while that of the nonlinear isotropic oscillator fluctuates decreasingly and degeneracies, completely removed. As the angular frequency is improved, expansion or increase is observed in the nonlinear and the inverse nonlinear cases in the radial wave functions. In the case of increase in the nonlinear term, saturation in the radial wave functions are observed in the inverted and non-inverted cases. For the effect of the increase in the magnetic field potential, initial decrease with saturation was observed in the both cases. The effect of the magnetic field and nonlinear term in both the nonlinear and inverted oscillators on the thermodynamic functions, attained saturation on the polyatomic molecules, while the effects of angular frequency is observed to enhance the thermodynamic functions. Generally, the contraction or saturation experienced with the increase of the magnetic field term is as a result of the small value of the electron charge, and the carbon (iv) oxide molecule is at higher potential than that of the water molecule.
We investigate the thermo-magnetic properties of the Attractive Radial Potential (ARP) for a particle moving in two-dimensional non-relativistic Schrödinger equation subjected to an external magnetic field and Aharonov-Bohm (AB) flux. Analytical solution for the energy eigenvalues and wavefunctions were obtained in a closed form via the Nikiforov-Uvarov Functional Analysis (NUFA) method. We derived the partition function by analyzing the system as a canonical ensemble to obtain the expressions for the thermodynamic quantities which include the free energy, entropy, specific heat capacity and internal energy. Our results show a decrease in the energy spectrum when the AB flux and magnetic fields are increased. At low temperatures, specific heat capacity shows a peak anomaly, while Helmholtz free energy and entropy exhibit temperature dependence. We also observed that AB flux and magnetic field influence both magnetization and magnetic susceptibility and exhibit both paramagnetic and diamagnetic behavior. The findings provide valuable insights into in molecular physics applications.
The solutions for a combination of the isotropic harmonic oscillator plus the inversely quadratic potentials and a combination of the pseudo-harmonic with inversely quadratic potentials has not been reported, though the individual potentials have been given attention. This study focuses on the solutions of the combination of the potentials, as stated above using the parametric Nikiforov–Uvarov (PNV) as the traditional technique to obtain the energy equations and their corresponding unnormalized radial wave functions. To deduce the application of these potentials, the expectation values, the uncertainty in the position and momentum, and the thermodynamic properties, such as the mean energy, entropy, heat capacity, and the free mean energy, are also calculated via the partition function. The result shows that the spectra for the PHIQ are higher than the spectra for the IHOIQ. It is also shown that the product of the uncertainties obeyed the Heisenberg uncertainty relation/principle. Finally, the thermal properties of the two potentials exhibit similar behaviours.
The exact solutions of the Schrodinger equations (SE) in the D-dimensional coordinate system have attracted the attention of many theoretical researchers in branches of quantum physics and quantum chemistry. The energy eigenvalues and the wave function are the solutions of the Schrodinger equation that implicitly represents the behavior of a quantum mechanical system. This study aimed to obtain the eigenvalues, wave functions, and thermodynamic properties of the 6-Dimensional Schrodinger equation under Double Ring-Shaped Oscillator (DRSO) and Manning-Rosen potential. The variable separation method was applied to reduce the one 6-Dimensional Schrodinger equation depending on radial and angular non-central potential into five onedimensional Schrodinger equations: one radial and five angular Schrodinger equations. Each of these onedimensional Schrodinger equations was solved using the SUSY QM method to obtain one eigenvalue and one wave function of the radial part, five eigenvalues, and five angular wave functions angular part. Some thermodynamic properties such, the vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆, were obtained using the radial energy equations. The results showed that except the 𝑛𝑙1, all increment of angular quantum number decreases the energy values. Increments of all potential parameter increase the energy values. Increment of angular quantum number and potentials parameter increases the amplitude and shifts the wave functions to the left. However, the increment of 𝑛𝑙1, 𝛼, 𝜎, and 𝜌 decrease the amplitude and shift wavefunctions to the right. Moreover, the vibrational mean energy 𝑈 and free energy 𝐹 increased as the increasing value of potentials parameters, where the ω parameter has the dominant effect than the other parameters. The vibrational specific heat 𝐶 and entropy 𝑆 affected only by the 𝜔 parameter, where 𝐶 and 𝑆 decreased as the increase of 𝜔.
Abstract We have calculated the effect of a magnetic field on the evolution of angular momentum eigenfunctions of a charged particle. An additional harmonic potential is supplemented to trap the wave packet. We find the probability density of the wave function is oscillating in the radial direction with a time period determined by the strength of the effective harmonic potential. When the magnetic field is along the z direction, if the initial wave function is an eigenfunction of L ˆ z , the probability density of the particle remains axis-symmetric. While for the case of an eigenfunction of L ˆ x , it is anisotropic in the x−y plane and rotates with a time period inverse proportional to the strength of the external magnetic field. We also extend the results in a phenomenological way to the case with an external magnetic field that varies harmonically in time.
In the present study, the improved screened Kratzer potential (ISKP) is investigated in the presence of external magnetic and Aharanov-Bohm (AB) fields within the framework of non-relativistic quantum mechanics. The Schrodinger equation is solved via the Nikiforov-Uvarov Functional Analysis (NUFA) method and the energy spectra and the corresponding wave function for the ISKP in the presence of external magnetic fields are obtained in a closed form. The obtained energy spectra are used to study some selected diatomic molecules. It is observed that the present of the magnetic and AB fields removes the degeneracy for different values of the control parameter. The thermodynamic and magnetic properties of the ISKP in the present of the magnetic and AB fields are also evaluated. The effects of the control potential parameter on the thermodynamic and magnetic properties of the selected diatomic molecules are discussed.
Abstract Attempts to understand the behaviour of various quantum systems are still become interest for many researchers in recent years. While experimentation is too complicated to be done, analytical mathematical approach is become a reliable choice. This approach is using potential functions or models as representations of the real quantum system, such as physical force interactions within a molecule. In this study, the new combination model of screened Kratzer and screened Eckart potentials is investigated under the non-relativistic quantum mechanics framework. These potentials are well-suited for modelling atomic interactions in diatomic molecules. The primary objective is to examine how these interactions are affected by the presence of external magnetic fields. Using a specific form of the Schro dinger equation, both the energy eigenfunction and the wave function have been derived. The Nikiforov-Uvarov Functional Analysis (NUFA) method was employed for this derivation. Additionally, several thermodynamic properties were obtained from the energy equation using numerical methods. Selected diatomic molecule data were used in the computations to demonstrate the effect of the magnetic field. A detailed analysis was performed through graphical representations, revealing that an increase in the magnetic field or the associated Aharonov-Bohm (AB) flux enhances and splits the energy spectra. The effects on thermodynamic properties vary across different molecules and conditions, but overall, the magnetic field increases the thermodynamic properties of the molecules.