Analytical solutions for the Schr\"odinger equation subjected to a deformable hyperbolic potential

TL;DR

This paper derives exact analytical solutions for the Schrödinger equation with a deformable hyperbolic tangent potential, providing insights into scattering states and potential energy comparisons.

Contribution

It presents explicit solutions in terms of hypergeometric functions for a deformable hyperbolic potential, expanding the analytical understanding of such quantum systems.

Findings

Analytical solutions in terms of hypergeometric functions for scattering states.

Comparison between deformable hyperbolic and abrupt step potentials.

Graphical illustrations of physical scenarios.

Abstract

In this work we discuss in detail the known solutions of the stationary Schr\"odinger equation subject to a deformable hyperbolic tangent potential exactly soluble $ V(x) = \frac {V_0} {2} (1+ \tanh (\delta x)) $. We find the analytical solutions in terms of Gauss hypergeometric functions for the scattering states with energy greater than the maximum value of the potential. We also discussed the case for the energy lower than the maximum and the similarities and differences with the abrupt step potential in both cases. We graphically illustrate the relevant physical situations to the problem.