Supersymmetry Approaches to the Bound States of the Generalized Woods–saxon Potential

Abstract

Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

Solutions for a generalized Woods–Saxon potential

“…This is one of the significant point in this article which employs another type of the Jacobi polynomials with 0 p τ ′ , unlike the works in [5,6], to initiate the present calculations.…”

“…For example, there is no explicit expression for n E for the Woods-Saxon potential, only a transcendent equation which has to be solved in order to obtain eigenvalues [4], although the corresponding wavefunction is known explicitly in terms of the hypergeometric fuction. Within this context, recently two works [5,6] involving different treatment techniques have been appeared in the literature to find an analytical solution for the potential of interest within the framework of non-relativistic physics. With the choice of the most frequently used type of the Jacobi polynomials with 0 = η γ , they have arrived at consistent expressions for the energy spectrum of an extended form of the Woods-Saxon potential.…”

“…Nevertheless, due to this wrong starting point leading to inevitably an improper hypergeometric function for the solutions, the analytical wavefunctions obtained in these works do not satisfy the required boundary condition in the asymptotic region. To remove this unphysical situation, Fakhri and his coworker [5] invoked a plausible weight function into the calculations. But this suggestion increases the calculational workload and leads to cumbersome procedures for the definition of the wavefunctions.…”

Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Potential Using Conventional NIKIFOROV-UVAROV Method

“…Nuclear shell model describes the interaction of nucleon with heavy nucleus [21][22][23]. Understanding the behavior of electrons in the valence shell has a major role in investigating metallic system [24].…”

Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential

“…Recently, the asymptotic iteration method (AIM) [15][16][17] an elegant, efficient technique to solve second-order homogeneous differential equations, has been the subject of extensive investigation in recent years, particularly when dealing withe non central potential. The Schrödinger equation has been investigated for several potentials as the Woods-Saxon potential [18][19][20], harmonic oscillator potential [21], Hulthén potential [22][23][24][25], Kratzer potential [26], generalized q-deformed Morse potential [27], modifed Woods-Saxon potential [28], Makarov potential [29], deformed Woods-Saxon Potential [30], Pseudoharmonic potential [31,32], Yukawa potential [33,34] and Eckart potential [35,36]. Very recently, the Schrödinger equation in generalized D dimensions for different potentials is getting more attention with the aim of generalizing the solutions to multidimensional space for many potentials [37][38][39][40][41][42][43][44][45][46].…”