Solvable Schrodinger Equations of Shape Invariant Potentials Having Superpotential W(x,A,B)=Atanh(px)+Btanh(6px)

TL;DR

This paper presents an exact solution to a new one-dimensional Schrödinger equation with a shape invariant potential, expanding the class of solvable quantum models using supersymmetric quantum mechanics.

Contribution

It introduces a novel shape invariant potential with superpotential involving hyperbolic functions and derives its exact solutions, which were previously unknown.

Findings

Exact eigenvalues and eigenfunctions derived

Potential applicable in nuclear physics and chemistry

Expands solvable models in quantum mechanics

Abstract

A new proposed one dimensional time independent Schr\"odinger equation is solved completely using shape invariance method. The corresponding potential is given by V_(x,A,B) =-A(sechpx)^2 - 6Bp(sech6px)^2+(tanhpx-6tanh6px)^2 with superpotential W(x,A,B) = Atanh(px)+Btanh(6px). We derive the exact solutions of the family of Schr\"odinger equations with the V_- potential partner using supersymmetric quantum mechanics technique of a superpotential having shape invariance property, and where the discrete spectrum and the corresponding eigenfunctions are determined exactly and in closed form. It is well-known that Schr\"odinger equations are challenging to solve in closed form, and only a few of them are known. Finding new equations with exact solutions is crucial in understanding the hidden physical properties near turning points where numerical methods fail in these vicinities. This result has potential applications in nuclear physics and chemistry where the antagonist forces have a prominent presence.