ANALYTICAL APPROXIMATIONS TO THE l-WAVE SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH A HYPERBOLIC POTENTIAL

Abstract

The bound-state solutions of the Schrödinger equation for a hyperbolic potential with the centrifugal term are presented approximately. It is shown that the solutions can be expressed by the hypergeometric function 2F1(a, b; c; z). To show the accuracy of our results, we calculate the energy levels numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other methods for short-range potential. Two special cases for l = 0 and σ = 1 are also studied briefly.

Approximate Analytical Solutions of the Dirac Equation with the Hyperbolic Potential in the Presence of the Spin Symmetry and Pseudo-spin Symmetry

“…By using a new improved approximation scheme to deal with the centrifugal term and pseudo-centrifugal term, we have solved approximately the Dirac equations with the relativistic hyperbolic potentials (9) and (44) in terms of the basic concept of the supersymmetric shape invariance formalism and the function analysis method. The energy eigenvalue equations and associated two-component spinors of the relativistic hyperbolic potentials (9) and (44) have been approximately obtained for the arbitrary spin-orbit quantum number κ.…”

“…By employing the same approximation scheme to deal with the centrifugal term, some authors [6,7] have studied approximately the bound state solutions of the Dirac equation and Klein-Gordon equation with the hyperbolic potential (1). Dong et al [8,9] also studied the bound state solutions of the Schrödinger equation with the potential (1) by employing the conventional approximation scheme suggested by Greene and Aldrich [10] to deal with the centrifugal term. Some authors have also used the conventional approximation scheme [10] to deal with the pseudo-centrifugal term, and studied the pseudospin symmetric solutions of the Dirac equations with the Eckart potential [11][12][13], Pöschl-Teller potential [14] and Manning-Rosen potential [15].…”

“…Some authors have also used the conventional approximation scheme [10] to deal with the pseudo-centrifugal term, and studied the pseudospin symmetric solutions of the Dirac equations with the Eckart potential [11][12][13], Pöschl-Teller potential [14] and Manning-Rosen potential [15]. The conventional approximation to the centrifugal term is a good approximation for the short-range potential, small α, but it is not a good approximation for the long-range potential, large α [8,9]. Recently, Wei et al [16,17] have also studied the bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equations with the Eckart potential and Manning-Rosen potential by using an alternative approximation scheme proposed by one of the present authors and collaborators [18] to deal with the centrifugal term.…”

Studies on the Bound-State Spectrum of Hyperbolic Potential

“…Later, arbitrary ℓ-state solutions are constructed in [10] by a proper approximation of the centrifugal term offering normalized functions in terms of generalized hypergeometric functions 2 F 1 (a, b; c; z). Arbitrary ℓ-wave solutions have also been suggested by an approximation [11] to the centrifugal term as 1 r 2 ≈ 4α 2 e −2αr (1−e −2αr ) 2 . Good-quality eigenvalues and eigenfunctions for general quantum numbers n, ℓ were presented by employing a Nikiforov-Uvarov approach [12][13][14], such that the approximate energy spectra and normalized total wave functions are represented in closed form through hypergeometric functions or Jacobi polynomials.…”

“…In another development [15], general eigensolutions are provided by means of an asymptotic iteration method in conjunction with a proper approximation to the centrifugal term. Very recently, non-relativistic ℓ-state solutions of N-dimensional Schrödinger equation with hyperbolic potential has been offered [16] in hyperspherical coordinates within the asymptotic iteration method along with an approximation to the centrifugal term along the lines of [11]. Relativistic bound-state solutions are also discussed by solving the Dirac equation with an aid of supersymmetric quantum mechanics and functional analysis method [17].…”