Arbitraryl-wave scattering state solutions of the Schrödinger equation for the Eckart potential

Abstract

The approximately analytical scattering state solutions of the l-wave Schrödinger equation for the Eckart potential are carried out by a proper approximation to the centrifugal term. The normalized radial wavefunctions of l-wave scattering states on the 'k/2π scale' are presented and the calculation formula of phase shifts is derived. It is interesting to find that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. We consider and verify two special cases: the l = 0 and the s-wave Hulthén potential.

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

“…Therefore, we must use a proper approximation for the centrifugal term similar to other authors. Unlike the following approximation used in the previous work [7][8][9][10][11][12][24][25][26],…”

“…For example, some authors have approximately solved the Schrödinger and Klein-Gordon equations for some po-tentials, these potentials include the Manning-Rosen potential [7], the Eckart potential [8], Hulthén potential [9], the Pöschl-Teller potential [10], etc. Based on previous work [7][8][9][10], we have applied a proper approximation to the centrifugal term to obtain approximately analytical −wave scattering solutions of the Schrödinger equation with the Manning-Rosen potential [11], the Eckart potential [12] and −wave bound solutions with the second Pöschl-Teller like potential [13] and the the hyperbolical potential [14] within the frame work of non-relativistic and relativistic quantum mechanics. The Manning-Rosen potential is an importantly solvable exponential-type potential describing diatomic molecules in quantum mechanics.…”

Bound States of the Klein-Gordon for Exponential-Type Potentials in D-Dimensions

“…Some authors [32][33][34][35][36][37][38] studied the analytical approximations to the bound state solutions of the Schrödinger equation with Eckart potential by using the usual existing approximation scheme proposed by Greene and Aldrich [53] for the centrifugal potential term. This approximation has also been used to study analytically the arbitrary l -wave scattering state solutions of the Schrödinger equation for the Eckart potential [54,55]. The same approximation scheme for the spinorbit coupling term has been used to study the spin symmetry and pseudospin symmetry analytical solutions of the Dirac equation with the Eckart potential using the AIM [56].…”

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

“…(4) cannot be solved analytically except for -states due to the centrifugal term. Therefore, we must use a proper approximation to the centrifugal term similar to previous literature [16,[18][19][20][21][22][23]. It is noted that for short potential range the following formula…”