Bound states of the $D$-dimensional Schr\"{o}dinger equation for the generalized Woods-Saxon potential

TL;DR

This paper derives approximate analytical solutions for the hyper-radial Schrödinger equation with the generalized Woods-Saxon potential in various dimensions, using the NU and SUSY QM methods, and applies them to neutron states in a $^{56}Fe$ nucleus.

Contribution

It introduces a combined approach using NU and SUSY QM methods to solve the generalized Woods-Saxon potential in multiple dimensions, providing explicit energy spectra and wave functions.

Findings

Derived energy eigenvalues and wave functions for various dimensions.

Calculated bound state energies for neutron in $^{56}Fe$ nucleus.

Identified energy spectra expressions for higher dimensions.

Abstract

In this paper, the approximate analitical solutions of the hyper-radial Schr\"{o}dinger equation are obtained for the generalized Wood-Saxon potential by implementing the Pekeris approximation to surmount the centrifugal term. The energy eigenvalues and corresponding hyper-radial wave functions are found for any angular momentum case via the Nikiforov-Uvarov (NU) and Supersymmetric quantum mechanics (SUSY QM) methods. Hence, the same expressions are obtained for the energy eigenvalues, and the expression of hyper-radial wave functions transformed each other is shown owing to these methods. Furthermore, a finite number energy spectrum depending on the depths of the potential well $V_{0}$ and $W$, the radial $n_{r}$ and $l$ orbital quantum numbers and parameters $D,a,R_{0}$ are also identified in detail. Finally, the bound state energies and the corresponding normalized hyper-radial wave functions for the neutron system of the a $^{56} Fe$ nucleus are calculated in $D=2$ and $D=3$, as well as the energy spectrum expressions of other highest dimensions are identified by using the energy spectrum of $D=2$ and $D=3$.