Analysis of Energy and Wave Functions and the Thermodynamics Properties of the 6-Dimensional Schrodinger Equation Under Double Ring-Shape Oscillator (Drso) and Manning-Rosen Potentials Using Susy Qm Method

Abstract

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.

Thermodynamic properties and superstatistics analysis of particle in the non-relativistic system under the influence of trigonometric scarf potential using Dong proper quantization

“…The exactly solvable relativistic and non-relativistic quantum systems have become interesting research areas to many researchers because they reveal all the important information to study quantum models. Some techniques had been used to solve the Schrödinger-like equations using various potentials [1][2], for example, the asymptotic iteration method (AIM) [3], the Nikiforov-Uvarov (NU) method [4], the Hypergeometric method [5][6], and the Laplace transform method [7]. The algebraic techniques that are related to the inspection of the Hamiltonian of a quantum system as in the supersymmetric quantum mechanics (SUSY QM) which is developed by the idea of shape invariant potential [8] and close to the factorization method [9], the SWKB method [10], and other methods which formed on the proper and exact quantization rule [11].…”