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Mathematical Methods in the Applied Sciences

Schrödinger equation based analytic assessment of thermal properties of confined electrons under vector and scalar fields

M. Eshghi, I. Ahmadi Azar, Sasan Soudi

Published Jun 11, 2021• 4 citations

Abstract

In this research, we have extracted the covariant form of the Schrödinger equation in the Riemannian manifold = ℝ3with a position‐dependent mass (PDM) distribution in the presence of magnetic fields for each point Then, we have solved the extracted nonrelativistic equation for a charged particle with a PDM in quantum system under the magnetic and Aharonov–Bohm (AB) flux fields and Yukawa‐like potential using the Frobenius series method. It is interesting that the obtained analytical solutions of the nonrelativistic equation have been expressed in terms of the biconfluent Heun function (BHF). Then, we have expanded the BHF as the power series and obtained a three‐term recurrence relation between the successive coefficients of the power expansion. To investigate the nontrivial solution of the power expansion, we need to apply an explicit continued fraction equation for the coefficients of the recurrence relation. This relation must be satisfied in order to assure the series convergence. At the continuation, we have found two different conditions using the submitted lemmas. Then, using the obtained conditions, we have calculated the eigenvalues and eigenfunctions of the particle in the arbitrary point of the manifold. Finally, the canonical formalism is applied to compute different thermodynamic variables for the case of the higher temperatures. Also, we have discussed the special states on the obtained results.

DOI: 10.1002/mma.7577

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In this research, we have extracted the covariant form of the Schrödinger equation in the Riemannian manifold = ℝ3with a position‐dependent mass (PDM) distribution in the presence of magnetic fields for each point Then, we have solved the extracted nonrelativistic equation for a charged particle with a PDM in quantum system under the magnetic and Aharonov–Bohm (AB) flux fields and Yukawa‐like potential using the Frobenius series method. It is interesting that the obtained analytical solutions of the nonrelativistic equation have been expressed in terms of the biconfluent Heun function (BHF). Then, we have expanded the BHF as the power series and obtained a three‐term recurrence relation between the successive coefficients of the power expansion. To investigate the nontrivial solution of the power expansion, we need to apply an explicit continued fraction equation for the coefficients of the recurrence relation. This relation must be satisfied in order to assure the series convergence. At the continuation, we have found two different conditions using the submitted lemmas. Then, using the obtained conditions, we have calculated the eigenvalues and eigenfunctions of the particle in the arbitrary point of the manifold. Finally, the canonical formalism is applied to compute different thermodynamic variables for the case of the higher temperatures. Also, we have discussed the special states on the obtained results.

We solve the Schrödinger equation with a position-dependent mass (PDM) charged particle interacted via the superposition of the Morse-plus-Coulomb potentials and is under the influence of external magnetic and Aharonov–Bohm (AB) flux fields. The nonrelativistic bound state energies together with their wave functions are calculated for two spatially-dependent mass distribution functions. We also study the thermal quantities of such a system. Further, the canonical formalism is used to compute various thermodynamic variables for second choosing mass by using the Gibbs formalism. We give plots for energy states as a function of various physical parameters. The behavior of the internal energy, specific heat, and entropy as functions of temperature and mass density parameter in the inverse-square mass case for different values of magnetic field are shown.

We need to solve a suitable exponential form of the position-dependent mass (PDM) Schrödinger equation with a charged particle placed in the Hulthen plus Coulomb-like potential field and under the actions of the external magnetic and Aharonov–Bohm (AB) flux fields. The bound state energies and their corresponding wave functions are calculated for the spatially-dependent mass distribution function of interest in physics. A few plots of some numerical results with respect to the energy are shown.

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 two-dimensional solution of the spinless Klein–Gordon (KG) equation for scalar–vector harmonic oscillator potentials with and without the presence of constant perpendicular magnetic and Aharonov–Bohm (AB) flux fields is studied within the asymptotic function analysis and Nikiforov–Uvarov (NU) method. The exact energy eigenvalues and normalized wave functions are analytically obtained in terms of potential parameters, magnetic field strength, AB flux field and magnetic quantum number. The results obtained by using different Larmor frequencies are compared with the results in the absence of both magnetic field (ωL = 0) and AB flux field (ξ = 0) case. Effects of external fields on the non-relativistic energy eigenvalues and wave functions solutions are also precisely presented.

Abstract The influence of external magnetic and Aharonov‐Bohm (AB) flux fields on the energy spectra and thermal properties of N2, NO, CO and H2 diatomic molecules are investigated in this study. The two‐dimensional Schrödinger equation is solved with the Varshni‐screened Coulomb potential (VSCP) model using the well‐known Nikiforov‐Uvarov (NU) method. We obtained the energy eigenvalues and wavefunctions of the system and calculated the energy levels of some diatomic molecules in the m‐arbitrary state in the presence and absence of external magnetic and AB flux fields. The results show that combining the magnetic and AB flux fields eliminates degeneracy. We analyzed and plotted the partition function and other thermal properties in the presence of a magnetic field, an AB field, and a temperature‐dependent term.

Abstract In this work, the position-dependent mass Schrödinger equation is solved with the Pöschl-Teller-like potential in the presence of magnetic and Aharonov–Bohm (AB) flux fields. The BenDaniel-Duke ambiguity parameter ordering is used to formulate the Hamiltonian operator for the system. An approximate analytical equation of the bound-state energy spectrum is obtained using the parametric Nikiforov-Uvarov solution technique along with a Pekeris-like approximation scheme. With the aid of the obtained equation for the energy levels, analytical formulas of magnetization and magnetic susceptibility at zero-temperature are derived and subsequently used to predict the physical properties of diatomic substances including the ground state H 2 , HCl, CO and LiH molecules. The expression for the bound-state-energy spectrum is used to generate numerical data for the molecules. The computed energy eigenvalues agree with the literature on diatomic molecules. The study revealed that in the absence of the external fields, the energy eigenvalues and magnetic susceptibility of the system are degenerate. However, with only a low intensity AB field, the degeneracy is completely eliminated from the energy states of the molecules.