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Journal of Physics Conference Series

Exact Solutions of Schrodinger Equation in Cylindrical Coordinates for Double Ring-Shaped Coulomb Oscillator Potential Using SUSY QM Method

Miftahul Ma’arif, Muhammad Abyan Naufal, Suparmi Suparmi, C. Cari

Published Jan 1, 2019• 12 citations
Publisher

Abstract

Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.

DOI
10.1088/1742-6596/1127/1/012005

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AIP conference proceedings•Jan 1, 2020•1 citation
Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.
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.
Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point [Formula: see text]. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections [Formula: see text] and [Formula: see text] by substituting [Formula: see text] and [Formula: see text].
In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.
Most of Schrdinger equations in quantum mechanics haven't analytic solutions except the problems of well with infinite depth,one-dimensional linear harmonic oscillator and Coulomb's potential as well as three-dimensional isotropic harmonic oscillator.These Schrdinger equations include the various equations in the non-spherical ring shaped oscillator potential that haven't analytic solutions. We can only solve these problems approximately in our research activities, but the solving process of these problems is often exceptional complex.The limitation handicaps us to lucubrate the idiographic problems.This paper adopts NU Method(A.F.Nikiforov-V.B.Uvarov Method) to solve the angular equation of the Schrdinger equation in the non-spherical ring shaped oscillator potential V(r,θ)=μω2r2/2+2α/(2μr2)+2βcosθ/(2μr2sin2θ).We decompose the Schrdinger equation in the non-spherical ring shaped oscillator potential step by step until we obtains the analytic solutions of the equation,thus we obtain the analytic solution of the angular eigenfunction,and simplify the process of solutions.Moreover,this paper adopts special functions in the radial equation at the same time to broaden our study in solving the Schrdinger equations.