Analytical solution of $D$ dimensional Schr\"odinger equation for Eckart potential with a new improved approximation in centrifugal term

TL;DR

This paper derives analytical solutions for the D-dimensional Schrödinger equation with Eckart potential using an improved approximation for the centrifugal term, enabling accurate energy and wavefunction calculations across dimensions.

Contribution

It introduces a new combined approximation for the centrifugal term, improving accuracy in solving the Schrödinger equation with Eckart potential in higher dimensions.

Findings

Solutions expressed in hypergeometric functions

Validated for arbitrary non-zero angular momentum states

Demonstrated accuracy across different dimensions and parameters

Abstract

Analytical solutions are presented for eigenvalues, eigenfunctions of {\color{red} D-dimensional Schrodinger equation having Eckart potential} within Nikiforov-Uvarov method. This uses a new, improved approximation for centrifugal term, from a combination of Greene-Aldrich and Pekeris approximations. Solutions are obtained in terms of hypergeometric functions. It facilitates an accurate representation in entire domain. Its validity is illustrated for energies in an arbitrary $\ell \neq 0$ quantum state. Results are compared for a chosen set of potential parameters in different dimensions. In short, a simple accurate approximation is offered for Eckart and other potentials in quantum mechanics, in higher dimension.