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

Abstract

A new proposed one dimensional time independent Schrödinger 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 Schrödinger 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 Schrödinger 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.

16 pages, 2 figures, 1 table

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