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arXiv:2509.12796 · Condensed Matter Physics

Generalised two-dimensional nonlinear oscillator with a position-dependent effective mass and the thermodynamic properties

S. E. Bokpe, F. A. Dossa, Gabriel Y. H. Avossevou

Published Sep 23, 2025• 0 citations
arXivHTMLPDF

Abstract

We investigate a two-dimensional nonlinear oscillator with a position-dependent effective mass in the framework of nonrelativistic quantum mechanics. Using the Nikiforov-Uvarov method, we obtain exact analytical expressions for the energy spectrum and wave functions. Based on the canonical partition function, we derive key thermodynamic quantities, including internal energy, specific heat, free energy, and entropy. Our results show that, unlike the one-dimensional case, where the specific heat is unaffected by the nonlinearity parameter k, the two-dimensional system exhibits a strong k−dependence. At high temperatures, the specific heat becomes temperature-independent for fixed values of k, in line with the Dulong–Petit law. However, these behaviors occur only for negative values of k. These findings highlight the impact of effective mass nonlinearity on macroscopic thermodynamic quantities and suggest that tuning the parameter k could serve as an effective strategy for enhancing the performance of quantum devices, including thermal machines and optoelectronic components.

DOI
10.5488/cmp.28.33701

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Abstract In this work, we find the exact solution of Schrödinger wave equation for position dependent mass with pseudoharmonic oscillator using extended Nikiforov-Uvarov method. We obtained the energy-eigen equation presented in a closed and compact form and used the result to study both superstatistics and thermodynamic properties by first determining the partition function of the system. The unnormalized wave function was obtained and expressed in terms of Confluent Heun function. Using the resulting energy eigen equation, the numerical computation were computed for varying masses for with fixed physical constant potential parameter λ. The numerical result shows that the bound state energies increase with quantum states but decreases with the dependent mass m(x). The thermodynamics and superstatistics plots are also reported.
A nonlinear oscillator with variable mass is studied in the approach of Supersymmetric quantum mechanics. Ladder operators in association with the shape invariance condition allowed us to find the coherent states of the system in the sense of Barut and Girardello. The statistical properties of these particular states are studied using the probability of distribution. In addition, the vibrational partition function is calculated. We see that the thermodynamic functions of the system such as mean energy and entropy depend on the nonlinearity parameter, except for the specific heat.
This work investigates a class of exactly solvable quantum systems described by a generalized Dunkl--Schrödinger equation with energy-dependent interactions. A two-parameter extension of the Dunkl differential operator is introduced, incorporating both the conventional reflection contribution and an additional derivative-reflection coupling. The resulting formalism leads to a modified Heisenberg algebra, from which the continuity equation, conserved probability current, and generalized normalization condition are derived for energy-dependent potentials. As an application, an energy-dependent harmonic oscillator is studied in detail. Exact analytical expressions for the eigenfunctions and energy spectrum are obtained in terms of associated Laguerre polynomials. The combined effects of the Wigner parameter, the deformation parameter, and the energy-dependence parameter are shown to produce parity splitting, nonlinear energy spectra, and nonuniform level spacing while preserving exact solvability. The thermodynamic properties of the model are then examined through the canonical partition function using the Euler--Maclaurin summation formula. Analytical expressions for the partition function, internal energy, entropy, and specific heat are derived and analyzed numerically. The results demonstrate that the generalized Dunkl deformation and energy-dependent interaction provide effective control over both the spectral and thermal characteristics of the system, extending the class of exactly solvable quantum models with reflection symmetry.
The evaluation of the specific heat of an open damped quantum system is a subtle issue. One possible route is based on the thermodynamic partition function which is the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. Being an open system this quantity then naturally must be interpreted as the change in the specific heat obtained as the difference between the specific heat of the heat bath coupled to the system degrees of freedom and the specific heat of the bath alone. While this difference may become negative, the involved specific heats themselves are always positive; thus, the known thermodynamic stability criteria are perfectly guaranteed. For a damped quantum harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Stylized minimal models containing a single oscillator heat bath are employed to elucidate the occurrence of the anomalous temperature dependence of the corresponding specific heat values. Moreover, we comment on the consequences for the interpretation of the density of states based on the thermal partition function.
The definition of the specific heat for a damped quantum system is a subtle issue. A possible approach is based on an effective partition function defined as the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. We argue that this quantity should be interpreted as the change of the specific heat of the heat bath when the system degrees of freedom are coupled to it. While this difference may become negative, the involved specific heats themselves are always positive and therefore thermodynamic stability is always guaranteed. For a damped harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Minimal models containing a single oscillator heat bath are employed to elucidate the temperature dependence of the difference of specific heats. Finally, we comment on the consequences for the interpretation of the density of states based on the effective partition function.
This work provides analytical study of a two-dimensional quantum harmonic oscillator (HO) coupled with a perpendicular magnetic field (B) in the x-direction and an inverse-square potential in the y-direction. The system’s Hamiltonian is exactly solved via variable separation, to obtain the energy levels and wavefunctions composed of Hermite and associated Laguerre polynomials. Fisher information (FI) and Shannon entropy (SE) for both position space (PS) and momentum space (MS) are fully examined under the magnetic field strength and anisotropic potential parameters. The canonical partition function is computed and thermodynamic quantities such as internal energy, entropy, and Gibbs free energy are evaluated. The result shows that the presence of magnetic field enhances spatial localization and reduces uncertainty in PS but has a revised effect in the MS. In the thermodynamic properties, there is asymmetric thermodynamic responses in the x- and y-directions due to the presence of the magnetic field and the inverse-square potential. This study highlights the deep relationship between quantum information theory and thermodynamics in low-dimensional systems with structured interactions.